Linear Algebra Examples

$[\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}]$

Step 1

Set up the formula to find the characteristic equation $p (λ)$ .

$p (λ) = determinant (A - λ I_{4})$

Step 2

The identity matrix or unit matrix of size $4$ is the $4 \times 4$ square matrix with ones on the main diagonal and zeros elsewhere.

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3

Substitute the known values into $p (λ) = determinant (A - λ I_{4})$ .

Tap for more steps...

Step 3.1

Substitute $[\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] - λ I_{4})$

Step 3.2

Substitute $[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$ for $I_{4}$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] - λ [\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}])$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply $- λ$ by each element of the matrix.

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2

Simplify each element in the matrix.

Tap for more steps...

Step 4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.4.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.5.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 λ & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.5.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ \cdot 1 & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.6

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & - λ \cdot 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.7.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.7.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.8.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.8.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.9.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.9.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.10

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.10.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 λ & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.10.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.11

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.12

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.12.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.12.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.13

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.13.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 λ & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.13.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & - λ \cdot 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.14

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.14.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 λ & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.14.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.15

Multiply $- λ \cdot 0$ .

Tap for more steps...

Step 4.1.2.15.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 λ & - λ \cdot 1 \end{array}])$

Step 4.1.2.15.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \cdot 1 \end{array}])$

Step 4.1.2.16

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \\ 4 & 4 & 4 & 4 \end{array}] + [\begin{array}{c} - λ & 0 & 0 & 0 \\ 0 & - λ & 0 & 0 \\ 0 & 0 & - λ & 0 \\ 0 & 0 & 0 & - λ \end{array}])$

Step 4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 + 0 & 4 + 0 & 4 + 0 \\ 4 + 0 & 4 - λ & 4 + 0 & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3

Simplify each element.

Tap for more steps...

Step 4.3.1

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 + 0 & 4 + 0 \\ 4 + 0 & 4 - λ & 4 + 0 & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.2

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 + 0 \\ 4 + 0 & 4 - λ & 4 + 0 & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.3

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 + 0 & 4 - λ & 4 + 0 & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.4

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 + 0 & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.5

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.6

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 & 4 \\ 4 + 0 & 4 + 0 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.7

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 & 4 \\ 4 & 4 + 0 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.8

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 & 4 \\ 4 & 4 & 4 - λ & 4 + 0 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.9

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 & 4 \\ 4 & 4 & 4 - λ & 4 \\ 4 + 0 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.10

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 & 4 \\ 4 & 4 & 4 - λ & 4 \\ 4 & 4 + 0 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.11

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 & 4 \\ 4 & 4 & 4 - λ & 4 \\ 4 & 4 & 4 + 0 & 4 - λ \end{array}]$

Step 4.3.12

Add $4$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 4 - λ & 4 & 4 & 4 \\ 4 & 4 - λ & 4 & 4 \\ 4 & 4 & 4 - λ & 4 \\ 4 & 4 & 4 & 4 - λ \end{array}]$

Step 5

Find the determinant.

Tap for more steps...

Step 5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

Tap for more steps...

Step 5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{array} |$

Step 5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 4 - λ & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} |$

Step 5.1.4

Multiply element $a_{11}$ by its cofactor.

$(4 - λ) | \begin{array}{c} 4 - λ & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} |$

Step 5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} |$

Step 5.1.6

Multiply element $a_{12}$ by its cofactor.

$- 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} |$

Step 5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} |$

Step 5.1.8

Multiply element $a_{13}$ by its cofactor.

$4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} |$

Step 5.1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.1.10

Multiply element $a_{14}$ by its cofactor.

$- 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.1.11

Add the terms together.

$p (λ) = (4 - λ) | \begin{array}{c} 4 - λ & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2

Evaluate $| \begin{array}{c} 4 - λ & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

Tap for more steps...

Step 5.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 5.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 4 - λ & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.2.1.4

Multiply element $a_{11}$ by its cofactor.

$(4 - λ) | \begin{array}{c} 4 - λ & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.2.1.6

Multiply element $a_{12}$ by its cofactor.

$- 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$

Step 5.2.1.8

Multiply element $a_{13}$ by its cofactor.

$4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$

Step 5.2.1.9

Add the terms together.

$p (λ) = (4 - λ) ((4 - λ) | \begin{array}{c} 4 - λ & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2

Evaluate $| \begin{array}{c} 4 - λ & 4 \\ 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.2.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) ((4 - λ) ((4 - λ) (4 - λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2

Simplify the determinant.

Tap for more steps...

Step 5.2.2.2.1

Simplify each term.

Tap for more steps...

Step 5.2.2.2.1.1

Expand $(4 - λ) (4 - λ)$ using the FOIL Method.

Tap for more steps...

Step 5.2.2.2.1.1.1

Apply the distributive property.

$p (λ) = (4 - λ) ((4 - λ) (4 (4 - λ) - λ (4 - λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.1.2

Apply the distributive property.

$p (λ) = (4 - λ) ((4 - λ) (4 \cdot 4 + 4 (- λ) - λ (4 - λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.1.3

Apply the distributive property.

$p (λ) = (4 - λ) ((4 - λ) (4 \cdot 4 + 4 (- λ) - λ \cdot 4 - λ (- λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 5.2.2.2.1.2.1

Simplify each term.

Tap for more steps...

Step 5.2.2.2.1.2.1.1

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) ((4 - λ) (16 + 4 (- λ) - λ \cdot 4 - λ (- λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2.1.2

Multiply $- 1$ by $4$ .

$p (λ) = (4 - λ) ((4 - λ) (16 - 4 λ - λ \cdot 4 - λ (- λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2.1.3

Multiply $4$ by $- 1$ .

$p (λ) = (4 - λ) ((4 - λ) (16 - 4 λ - 4 λ - λ (- λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = (4 - λ) ((4 - λ) (16 - 4 λ - 4 λ - 1 \cdot - 1 λ \cdot λ - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2.1.5

Multiply $λ$ by $λ$ by adding the exponents.

Tap for more steps...

Step 5.2.2.2.1.2.1.5.1

Move $λ$ .

$p (λ) = (4 - λ) ((4 - λ) (16 - 4 λ - 4 λ - 1 \cdot - 1 (λ \cdot λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = (4 - λ) ((4 - λ) (16 - 4 λ - 4 λ - 1 \cdot - 1 λ^{2} - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = (4 - λ) ((4 - λ) (16 - 4 λ - 4 λ + 1 λ^{2} - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = (4 - λ) ((4 - λ) (16 - 4 λ - 4 λ + λ^{2} - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.2.2

Subtract $4 λ$ from $- 4 λ$ .

$p (λ) = (4 - λ) ((4 - λ) (16 - 8 λ + λ^{2} - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.1.3

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) ((4 - λ) (16 - 8 λ + λ^{2} - 16) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.2

Combine the opposite terms in $16 - 8 λ + λ^{2} - 16$ .

Tap for more steps...

Step 5.2.2.2.2.1

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) ((4 - λ) (- 8 λ + λ^{2} + 0) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.2.2

Add $- 8 λ + λ^{2}$ and $0$ .

$p (λ) = (4 - λ) ((4 - λ) (- 8 λ + λ^{2}) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.2.2.3

Reorder $- 8 λ$ and $λ^{2}$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.3

Evaluate $| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.2.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (4 (4 - λ) - 4 \cdot 4) + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.3.2

Simplify the determinant.

Tap for more steps...

Step 5.2.3.2.1

Simplify each term.

Tap for more steps...

Step 5.2.3.2.1.1

Apply the distributive property.

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (4 \cdot 4 + 4 (- λ) - 4 \cdot 4) + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.3.2.1.2

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (16 + 4 (- λ) - 4 \cdot 4) + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.3.2.1.3

Multiply $- 1$ by $4$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (16 - 4 λ - 4 \cdot 4) + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.3.2.1.4

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (16 - 4 λ - 16) + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.3.2.2

Combine the opposite terms in $16 - 4 λ - 16$ .

Tap for more steps...

Step 5.2.3.2.2.1

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ + 0) + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.3.2.2.2

Add $- 4 λ$ and $0$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.4

Evaluate $| \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$ .

Tap for more steps...

Step 5.2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (4 \cdot 4 - 4 (4 - λ))) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.4.2

Simplify the determinant.

Tap for more steps...

Step 5.2.4.2.1

Simplify each term.

Tap for more steps...

Step 5.2.4.2.1.1

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (16 - 4 (4 - λ))) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.4.2.1.2

Apply the distributive property.

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (16 - 4 \cdot 4 - 4 (- λ))) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.4.2.1.3

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (16 - 16 - 4 (- λ))) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.4.2.1.4

Multiply $- 1$ by $- 4$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (16 - 16 + 4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.4.2.2

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (0 + 4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.4.2.3

Add $0$ and $4 λ$ .

$p (λ) = (4 - λ) ((4 - λ) (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5

Simplify the determinant.

Tap for more steps...

Step 5.2.5.1

Simplify each term.

Tap for more steps...

Step 5.2.5.1.1

Expand $(4 - λ) (λ^{2} - 8 λ)$ using the FOIL Method.

Tap for more steps...

Step 5.2.5.1.1.1

Apply the distributive property.

$p (λ) = (4 - λ) (4 (λ^{2} - 8 λ) - λ (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.1.2

Apply the distributive property.

$p (λ) = (4 - λ) (4 λ^{2} + 4 (- 8 λ) - λ (λ^{2} - 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.1.3

Apply the distributive property.

$p (λ) = (4 - λ) (4 λ^{2} + 4 (- 8 λ) - λ \cdot λ^{2} - λ (- 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2

Simplify and combine like terms.

Tap for more steps...

Step 5.2.5.1.2.1

Simplify each term.

Tap for more steps...

Step 5.2.5.1.2.1.1

Multiply $- 8$ by $4$ .

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - λ \cdot λ^{2} - λ (- 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.1.2

Multiply $λ$ by $λ^{2}$ by adding the exponents.

Tap for more steps...

Step 5.2.5.1.2.1.2.1

Move $λ^{2}$ .

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - (λ^{2} λ) - λ (- 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.1.2.2

Multiply $λ^{2}$ by $λ$ .

Tap for more steps...

Step 5.2.5.1.2.1.2.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - (λ^{2} λ^{1}) - λ (- 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - λ^{2 + 1} - λ (- 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.1.2.3

Add $2$ and $1$ .

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - λ^{3} - λ (- 8 λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.1.3

Rewrite using the commutative property of multiplication.

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - λ^{3} - 1 \cdot - 8 λ \cdot λ - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.1.4

Multiply $λ$ by $λ$ by adding the exponents.

Tap for more steps...

Step 5.2.5.1.2.1.4.1

Move $λ$ .

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - λ^{3} - 1 \cdot - 8 (λ \cdot λ) - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.1.4.2

Multiply $λ$ by $λ$ .

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - λ^{3} - 1 \cdot - 8 λ^{2} - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.1.5

Multiply $- 1$ by $- 8$ .

$p (λ) = (4 - λ) (4 λ^{2} - 32 λ - λ^{3} + 8 λ^{2} - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.2.2

Add $4 λ^{2}$ and $8 λ^{2}$ .

$p (λ) = (4 - λ) (12 λ^{2} - 32 λ - λ^{3} - 4 (- 4 λ) + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.3

Multiply $- 4$ by $- 4$ .

$p (λ) = (4 - λ) (12 λ^{2} - 32 λ - λ^{3} + 16 λ + 4 (4 λ)) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.1.4

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (12 λ^{2} - 32 λ - λ^{3} + 16 λ + 16 λ) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.2

Add $- 32 λ$ and $16 λ$ .

$p (λ) = (4 - λ) (12 λ^{2} - λ^{3} - 16 λ + 16 λ) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.3

Combine the opposite terms in $12 λ^{2} - λ^{3} - 16 λ + 16 λ$ .

Tap for more steps...

Step 5.2.5.3.1

Add $- 16 λ$ and $16 λ$ .

$p (λ) = (4 - λ) (12 λ^{2} - λ^{3} + 0) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.3.2

Add $12 λ^{2} - λ^{3}$ and $0$ .

$p (λ) = (4 - λ) (12 λ^{2} - λ^{3}) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.2.5.4

Reorder $12 λ^{2}$ and $- λ^{3}$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 | \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3

Evaluate $| \begin{array}{c} 4 & 4 & 4 \\ 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.3.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $2$ by its cofactor and add.

Tap for more steps...

Step 5.3.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 5.3.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.3.1.3

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.3.1.4

Multiply element $a_{21}$ by its cofactor.

$- 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.3.1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.3.1.6

Multiply element $a_{22}$ by its cofactor.

$(4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.3.1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$

Step 5.3.1.8

Multiply element $a_{23}$ by its cofactor.

$- 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$

Step 5.3.1.9

Add the terms together.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.2

Evaluate $| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.3.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (4 (4 - λ) - 4 \cdot 4) + (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.2.2

Simplify the determinant.

Tap for more steps...

Step 5.3.2.2.1

Simplify each term.

Tap for more steps...

Step 5.3.2.2.1.1

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (4 \cdot 4 + 4 (- λ) - 4 \cdot 4) + (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.2.2.1.2

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (16 + 4 (- λ) - 4 \cdot 4) + (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.2.2.1.3

Multiply $- 1$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (16 - 4 λ - 4 \cdot 4) + (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.2.2.1.4

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (16 - 4 λ - 16) + (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.2.2.2

Combine the opposite terms in $16 - 4 λ - 16$ .

Tap for more steps...

Step 5.3.2.2.2.1

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ + 0) + (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.2.2.2.2

Add $- 4 λ$ and $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.3

Evaluate $| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.3.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (4 (4 - λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.3.2

Simplify the determinant.

Tap for more steps...

Step 5.3.3.2.1

Simplify each term.

Tap for more steps...

Step 5.3.3.2.1.1

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (4 \cdot 4 + 4 (- λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.3.2.1.2

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (16 + 4 (- λ) - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.3.2.1.3

Multiply $- 1$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (16 - 4 λ - 4 \cdot 4) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.3.2.1.4

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (16 - 4 λ - 16) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.3.2.2

Combine the opposite terms in $16 - 4 λ - 16$ .

Tap for more steps...

Step 5.3.3.2.2.1

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (- 4 λ + 0) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.3.2.2.2

Add $- 4 λ$ and $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (- 4 λ) - 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.4

Evaluate $| \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$ .

Tap for more steps...

Step 5.3.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (- 4 λ) - 4 (4 \cdot 4 - 4 \cdot 4)) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.4.2

Simplify the determinant.

Tap for more steps...

Step 5.3.4.2.1

Simplify each term.

Tap for more steps...

Step 5.3.4.2.1.1

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (- 4 λ) - 4 (16 - 4 \cdot 4)) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.4.2.1.2

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (- 4 λ) - 4 (16 - 16)) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.4.2.2

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (- 4 (- 4 λ) + (4 - λ) (- 4 λ) - 4 \cdot 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5

Simplify the determinant.

Tap for more steps...

Step 5.3.5.1

Simplify each term.

Tap for more steps...

Step 5.3.5.1.1

Multiply $- 4$ by $- 4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ + (4 - λ) (- 4 λ) - 4 \cdot 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.1.2

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ + 4 (- 4 λ) - λ (- 4 λ) - 4 \cdot 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.1.3

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ - 16 λ - λ (- 4 λ) - 4 \cdot 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ - 16 λ - 1 \cdot - 4 λ \cdot λ - 4 \cdot 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.1.5

Simplify each term.

Tap for more steps...

Step 5.3.5.1.5.1

Multiply $λ$ by $λ$ by adding the exponents.

Tap for more steps...

Step 5.3.5.1.5.1.1

Move $λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ - 16 λ - 1 \cdot - 4 (λ \cdot λ) - 4 \cdot 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.1.5.1.2

Multiply $λ$ by $λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ - 16 λ - 1 \cdot - 4 λ^{2} - 4 \cdot 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.1.5.2

Multiply $- 1$ by $- 4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ - 16 λ + 4 λ^{2} - 4 \cdot 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.1.6

Multiply $- 4$ by $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ - 16 λ + 4 λ^{2} + 0) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.2

Combine the opposite terms in $16 λ - 16 λ + 4 λ^{2} + 0$ .

Tap for more steps...

Step 5.3.5.2.1

Add $16 λ - 16 λ + 4 λ^{2}$ and $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (16 λ - 16 λ + 4 λ^{2}) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.2.2

Subtract $16 λ$ from $16 λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (0 + 4 λ^{2}) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.3.5.3

Add $0$ and $4 λ^{2}$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} | - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4

Evaluate $| \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 \\ 4 & 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.4.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

Tap for more steps...

Step 5.4.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 5.4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.4.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.4.1.4

Multiply element $a_{11}$ by its cofactor.

$4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.4.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.4.1.6

Multiply element $a_{12}$ by its cofactor.

$- (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$

Step 5.4.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$

Step 5.4.1.8

Multiply element $a_{13}$ by its cofactor.

$4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$

Step 5.4.1.9

Add the terms together.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | - (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.2

Evaluate $| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (4 (4 - λ) - 4 \cdot 4) - (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.2.2

Simplify the determinant.

Tap for more steps...

Step 5.4.2.2.1

Simplify each term.

Tap for more steps...

Step 5.4.2.2.1.1

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (4 \cdot 4 + 4 (- λ) - 4 \cdot 4) - (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.2.2.1.2

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (16 + 4 (- λ) - 4 \cdot 4) - (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.2.2.1.3

Multiply $- 1$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (16 - 4 λ - 4 \cdot 4) - (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.2.2.1.4

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (16 - 4 λ - 16) - (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.2.2.2

Combine the opposite terms in $16 - 4 λ - 16$ .

Tap for more steps...

Step 5.4.2.2.2.1

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ + 0) - (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.2.2.2.2

Add $- 4 λ$ and $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) | \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.3

Evaluate $| \begin{array}{c} 4 & 4 \\ 4 & 4 - λ \end{array} |$ .

Tap for more steps...

Step 5.4.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (4 (4 - λ) - 4 \cdot 4) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.3.2

Simplify the determinant.

Tap for more steps...

Step 5.4.3.2.1

Simplify each term.

Tap for more steps...

Step 5.4.3.2.1.1

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (4 \cdot 4 + 4 (- λ) - 4 \cdot 4) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.3.2.1.2

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (16 + 4 (- λ) - 4 \cdot 4) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.3.2.1.3

Multiply $- 1$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (16 - 4 λ - 4 \cdot 4) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.3.2.1.4

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (16 - 4 λ - 16) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.3.2.2

Combine the opposite terms in $16 - 4 λ - 16$ .

Tap for more steps...

Step 5.4.3.2.2.1

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (- 4 λ + 0) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.3.2.2.2

Add $- 4 λ$ and $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (- 4 λ) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.4

Evaluate $| \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$ .

Tap for more steps...

Step 5.4.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (- 4 λ) + 4 (4 \cdot 4 - 4 \cdot 4)) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.4.2

Simplify the determinant.

Tap for more steps...

Step 5.4.4.2.1

Simplify each term.

Tap for more steps...

Step 5.4.4.2.1.1

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (- 4 λ) + 4 (16 - 4 \cdot 4)) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.4.2.1.2

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (- 4 λ) + 4 (16 - 16)) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.4.2.2

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (4 (- 4 λ) - (4 - λ) (- 4 λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5

Simplify the determinant.

Tap for more steps...

Step 5.4.5.1

Simplify each term.

Tap for more steps...

Step 5.4.5.1.1

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ - (4 - λ) (- 4 λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.2

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + (- 1 \cdot 4 - - λ) (- 4 λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.3

Multiply $- 1$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + (- 4 - - λ) (- 4 λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.4

Multiply $- - λ$ .

Tap for more steps...

Step 5.4.5.1.4.1

Multiply $- 1$ by $- 1$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + (- 4 + 1 λ) (- 4 λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.4.2

Multiply $λ$ by $1$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + (- 4 + λ) (- 4 λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.5

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ - 4 (- 4 λ) + λ (- 4 λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.6

Multiply $- 4$ by $- 4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + 16 λ + λ (- 4 λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.7

Rewrite using the commutative property of multiplication.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + 16 λ - 4 λ \cdot λ + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.8

Multiply $λ$ by $λ$ by adding the exponents.

Tap for more steps...

Step 5.4.5.1.8.1

Move $λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + 16 λ - 4 (λ \cdot λ) + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.8.2

Multiply $λ$ by $λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + 16 λ - 4 λ^{2} + 4 \cdot 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.1.9

Multiply $4$ by $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + 16 λ - 4 λ^{2} + 0) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.2

Combine the opposite terms in $- 16 λ + 16 λ - 4 λ^{2} + 0$ .

Tap for more steps...

Step 5.4.5.2.1

Add $- 16 λ + 16 λ - 4 λ^{2}$ and $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 16 λ + 16 λ - 4 λ^{2}) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.2.2

Add $- 16 λ$ and $16 λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (0 - 4 λ^{2}) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.4.5.3

Subtract $4 λ^{2}$ from $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 | \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$

Step 5.5

Evaluate $| \begin{array}{c} 4 & 4 - λ & 4 \\ 4 & 4 & 4 - λ \\ 4 & 4 & 4 \end{array} |$ .

Tap for more steps...

Step 5.5.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

Tap for more steps...

Step 5.5.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 5.5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.5.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$

Step 5.5.1.4

Multiply element $a_{11}$ by its cofactor.

$4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$

Step 5.5.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$

Step 5.5.1.6

Multiply element $a_{12}$ by its cofactor.

$- (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$

Step 5.5.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$

Step 5.5.1.8

Multiply element $a_{13}$ by its cofactor.

$4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$

Step 5.5.1.9

Add the terms together.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | - (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.2

Evaluate $| \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$ .

Tap for more steps...

Step 5.5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 \cdot 4 - 4 (4 - λ)) - (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.2.2

Simplify the determinant.

Tap for more steps...

Step 5.5.2.2.1

Simplify each term.

Tap for more steps...

Step 5.5.2.2.1.1

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (16 - 4 (4 - λ)) - (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.2.2.1.2

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (16 - 4 \cdot 4 - 4 (- λ)) - (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.2.2.1.3

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (16 - 16 - 4 (- λ)) - (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.2.2.1.4

Multiply $- 1$ by $- 4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (16 - 16 + 4 λ) - (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.2.2.2

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (0 + 4 λ) - (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.2.2.3

Add $0$ and $4 λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) | \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} | + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.3

Evaluate $| \begin{array}{c} 4 & 4 - λ \\ 4 & 4 \end{array} |$ .

Tap for more steps...

Step 5.5.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (4 \cdot 4 - 4 (4 - λ)) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.3.2

Simplify the determinant.

Tap for more steps...

Step 5.5.3.2.1

Simplify each term.

Tap for more steps...

Step 5.5.3.2.1.1

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (16 - 4 (4 - λ)) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.3.2.1.2

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (16 - 4 \cdot 4 - 4 (- λ)) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.3.2.1.3

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (16 - 16 - 4 (- λ)) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.3.2.1.4

Multiply $- 1$ by $- 4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (16 - 16 + 4 λ) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.3.2.2

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (0 + 4 λ) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.3.2.3

Add $0$ and $4 λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (4 λ) + 4 | \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |)$

Step 5.5.4

Evaluate $| \begin{array}{c} 4 & 4 \\ 4 & 4 \end{array} |$ .

Tap for more steps...

Step 5.5.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (4 λ) + 4 (4 \cdot 4 - 4 \cdot 4))$

Step 5.5.4.2

Simplify the determinant.

Tap for more steps...

Step 5.5.4.2.1

Simplify each term.

Tap for more steps...

Step 5.5.4.2.1.1

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (4 λ) + 4 (16 - 4 \cdot 4))$

Step 5.5.4.2.1.2

Multiply $- 4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (4 λ) + 4 (16 - 16))$

Step 5.5.4.2.2

Subtract $16$ from $16$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 (4 λ) - (4 - λ) (4 λ) + 4 \cdot 0)$

Step 5.5.5

Simplify the determinant.

Tap for more steps...

Step 5.5.5.1

Simplify each term.

Tap for more steps...

Step 5.5.5.1.1

Multiply $4$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ - (4 - λ) (4 λ) + 4 \cdot 0)$

Step 5.5.5.1.2

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ + (- 1 \cdot 4 - - λ) (4 λ) + 4 \cdot 0)$

Step 5.5.5.1.3

Multiply $- 1$ by $4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ + (- 4 - - λ) (4 λ) + 4 \cdot 0)$

Step 5.5.5.1.4

Multiply $- - λ$ .

Tap for more steps...

Step 5.5.5.1.4.1

Multiply $- 1$ by $- 1$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ + (- 4 + 1 λ) (4 λ) + 4 \cdot 0)$

Step 5.5.5.1.4.2

Multiply $λ$ by $1$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ + (- 4 + λ) (4 λ) + 4 \cdot 0)$

Step 5.5.5.1.5

Apply the distributive property.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ - 4 (4 λ) + λ (4 λ) + 4 \cdot 0)$

Step 5.5.5.1.6

Multiply $4$ by $- 4$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ - 16 λ + λ (4 λ) + 4 \cdot 0)$

Step 5.5.5.1.7

Rewrite using the commutative property of multiplication.

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ - 16 λ + 4 λ \cdot λ + 4 \cdot 0)$

Step 5.5.5.1.8

Multiply $λ$ by $λ$ by adding the exponents.

Tap for more steps...

Step 5.5.5.1.8.1

Move $λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ - 16 λ + 4 (λ \cdot λ) + 4 \cdot 0)$

Step 5.5.5.1.8.2

Multiply $λ$ by $λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ - 16 λ + 4 λ^{2} + 4 \cdot 0)$

Step 5.5.5.1.9

Multiply $4$ by $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ - 16 λ + 4 λ^{2} + 0)$

Step 5.5.5.2

Combine the opposite terms in $16 λ - 16 λ + 4 λ^{2} + 0$ .

Tap for more steps...

Step 5.5.5.2.1

Add $16 λ - 16 λ + 4 λ^{2}$ and $0$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (16 λ - 16 λ + 4 λ^{2})$

Step 5.5.5.2.2

Subtract $16 λ$ from $16 λ$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (0 + 4 λ^{2})$

Step 5.5.5.3

Add $0$ and $4 λ^{2}$ .

$p (λ) = (4 - λ) (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6

Simplify the determinant.

Tap for more steps...

Step 5.6.1

Simplify each term.

Tap for more steps...

Step 5.6.1.1

Expand $(4 - λ) (- λ^{3} + 12 λ^{2})$ using the FOIL Method.

Tap for more steps...

Step 5.6.1.1.1

Apply the distributive property.

$p (λ) = 4 (- λ^{3} + 12 λ^{2}) - λ (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.1.2

Apply the distributive property.

$p (λ) = 4 (- λ^{3}) + 4 (12 λ^{2}) - λ (- λ^{3} + 12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.1.3

Apply the distributive property.

$p (λ) = 4 (- λ^{3}) + 4 (12 λ^{2}) - λ (- λ^{3}) - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2

Simplify and combine like terms.

Tap for more steps...

Step 5.6.1.2.1

Simplify each term.

Tap for more steps...

Step 5.6.1.2.1.1

Multiply $- 1$ by $4$ .

$p (λ) = - 4 λ^{3} + 4 (12 λ^{2}) - λ (- λ^{3}) - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.2

Multiply $12$ by $4$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} - λ (- λ^{3}) - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.3

Rewrite using the commutative property of multiplication.

$p (λ) = - 4 λ^{3} + 48 λ^{2} - 1 \cdot - 1 λ \cdot λ^{3} - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.4

Multiply $λ$ by $λ^{3}$ by adding the exponents.

Tap for more steps...

Step 5.6.1.2.1.4.1

Move $λ^{3}$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} - 1 \cdot - 1 (λ^{3} λ) - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.4.2

Multiply $λ^{3}$ by $λ$ .

Tap for more steps...

Step 5.6.1.2.1.4.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} - 1 \cdot - 1 (λ^{3} λ^{1}) - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - 4 λ^{3} + 48 λ^{2} - 1 \cdot - 1 λ^{3 + 1} - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.4.3

Add $3$ and $1$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} - 1 \cdot - 1 λ^{4} - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.5

Multiply $- 1$ by $- 1$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} + 1 λ^{4} - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.6

Multiply $λ^{4}$ by $1$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} + λ^{4} - λ (12 λ^{2}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.7

Rewrite using the commutative property of multiplication.

$p (λ) = - 4 λ^{3} + 48 λ^{2} + λ^{4} - 1 \cdot 12 λ \cdot λ^{2} - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.8

Multiply $λ$ by $λ^{2}$ by adding the exponents.

Tap for more steps...

Step 5.6.1.2.1.8.1

Move $λ^{2}$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} + λ^{4} - 1 \cdot 12 (λ^{2} λ) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.8.2

Multiply $λ^{2}$ by $λ$ .

Tap for more steps...

Step 5.6.1.2.1.8.2.1

Raise $λ$ to the power of $1$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} + λ^{4} - 1 \cdot 12 (λ^{2} λ^{1}) - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$p (λ) = - 4 λ^{3} + 48 λ^{2} + λ^{4} - 1 \cdot 12 λ^{2 + 1} - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.8.3

Add $2$ and $1$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} + λ^{4} - 1 \cdot 12 λ^{3} - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.1.9

Multiply $- 1$ by $12$ .

$p (λ) = - 4 λ^{3} + 48 λ^{2} + λ^{4} - 12 λ^{3} - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.2.2

Subtract $12 λ^{3}$ from $- 4 λ^{3}$ .

$p (λ) = - 16 λ^{3} + 48 λ^{2} + λ^{4} - 4 (4 λ^{2}) + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.3

Multiply $4$ by $- 4$ .

$p (λ) = - 16 λ^{3} + 48 λ^{2} + λ^{4} - 16 λ^{2} + 4 (- 4 λ^{2}) - 4 (4 λ^{2})$

Step 5.6.1.4

Multiply $- 4$ by $4$ .

$p (λ) = - 16 λ^{3} + 48 λ^{2} + λ^{4} - 16 λ^{2} - 16 λ^{2} - 4 (4 λ^{2})$

Step 5.6.1.5

Multiply $4$ by $- 4$ .

$p (λ) = - 16 λ^{3} + 48 λ^{2} + λ^{4} - 16 λ^{2} - 16 λ^{2} - 16 λ^{2}$

Step 5.6.2

Subtract $16 λ^{2}$ from $48 λ^{2}$ .

$p (λ) = - 16 λ^{3} + λ^{4} + 32 λ^{2} - 16 λ^{2} - 16 λ^{2}$

Step 5.6.3

Subtract $16 λ^{2}$ from $32 λ^{2}$ .

$p (λ) = - 16 λ^{3} + λ^{4} + 16 λ^{2} - 16 λ^{2}$

Step 5.6.4

Combine the opposite terms in $- 16 λ^{3} + λ^{4} + 16 λ^{2} - 16 λ^{2}$ .

Tap for more steps...

Step 5.6.4.1

Subtract $16 λ^{2}$ from $16 λ^{2}$ .

$p (λ) = - 16 λ^{3} + λ^{4} + 0$

Step 5.6.4.2

Add $- 16 λ^{3} + λ^{4}$ and $0$ .

$p (λ) = - 16 λ^{3} + λ^{4}$

Step 5.6.5

Reorder $- 16 λ^{3}$ and $λ^{4}$ .

$p (λ) = λ^{4} - 16 λ^{3}$

Step 6

Set the characteristic polynomial equal to $0$ to find the eigenvalues $λ$ .

$λ^{4} - 16 λ^{3} = 0$

Step 7

Solve for $λ$ .

Tap for more steps...

Step 7.1

Factor $λ^{3}$ out of $λ^{4} - 16 λ^{3}$ .

Tap for more steps...

Step 7.1.1

Factor $λ^{3}$ out of $λ^{4}$ .

$λ^{3} λ - 16 λ^{3} = 0$

Step 7.1.2

Factor $λ^{3}$ out of $- 16 λ^{3}$ .

$λ^{3} λ + λ^{3} \cdot - 16 = 0$

Step 7.1.3

Factor $λ^{3}$ out of $λ^{3} λ + λ^{3} \cdot - 16$ .

$λ^{3} (λ - 16) = 0$

Step 7.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$λ^{3} = 0$

$λ - 16 = 0$

Step 7.3

Set $λ^{3}$ equal to $0$ and solve for $λ$ .

Tap for more steps...

Step 7.3.1

Set $λ^{3}$ equal to $0$ .

$λ^{3} = 0$

Step 7.3.2

Solve $λ^{3} = 0$ for $λ$ .

Tap for more steps...

Step 7.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$λ = \sqrt[3]{0}$

Step 7.3.2.2

Simplify $\sqrt[3]{0}$ .

Tap for more steps...

Step 7.3.2.2.1

Rewrite $0$ as $0^{3}$ .

$λ = \sqrt[3]{0^{3}}$

Step 7.3.2.2.2

Pull terms out from under the radical, assuming real numbers.

$λ = 0$

Step 7.4

Set $λ - 16$ equal to $0$ and solve for $λ$ .

Tap for more steps...

Step 7.4.1

Set $λ - 16$ equal to $0$ .

$λ - 16 = 0$

Step 7.4.2

Add $16$ to both sides of the equation.

$λ = 16$

Step 7.5

The final solution is all the values that make $λ^{3} (λ - 16) = 0$ true.

$λ = 0, 16$