Linear Algebra Examples

$[\begin{array}{c} 1 - x & 1 & - 2 \\ - 1 & 2 - x & 1 \\ 0 & 1 & - 1 - x \end{array}]$

Step 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 column $1$ by its cofactor and add.

Tap for more steps...

Step 1.1

Consider the corresponding sign chart.

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

Step 1.2

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

Step 1.3

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

$| \begin{array}{c} 2 - x & 1 \\ 1 & - 1 - x \end{array} |$

Step 1.4

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

$(1 - x) | \begin{array}{c} 2 - x & 1 \\ 1 & - 1 - x \end{array} |$

Step 1.5

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

$| \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} |$

Step 1.6

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

$1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} |$

Step 1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 1 & - 2 \\ 2 - x & 1 \end{array} |$

Step 1.8

Multiply element $a_{31}$ by its cofactor.

$0 | \begin{array}{c} 1 & - 2 \\ 2 - x & 1 \end{array} |$

Step 1.9

Add the terms together.

$(1 - x) | \begin{array}{c} 2 - x & 1 \\ 1 & - 1 - x \end{array} | + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0 | \begin{array}{c} 1 & - 2 \\ 2 - x & 1 \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} 1 & - 2 \\ 2 - x & 1 \end{array} |$ .

$(1 - x) | \begin{array}{c} 2 - x & 1 \\ 1 & - 1 - x \end{array} | + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3

Evaluate $| \begin{array}{c} 2 - x & 1 \\ 1 & - 1 - x \end{array} |$ .

Tap for more steps...

Step 3.1

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

$(1 - x) ((2 - x) (- 1 - x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2

Simplify the determinant.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Expand $(2 - x) (- 1 - x)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.1.1

Apply the distributive property.

$(1 - x) (2 (- 1 - x) - x (- 1 - x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.1.2

Apply the distributive property.

$(1 - x) (2 \cdot - 1 + 2 (- x) - x (- 1 - x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.1.3

Apply the distributive property.

$(1 - x) (2 \cdot - 1 + 2 (- x) - x \cdot - 1 - x (- x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.2.1.1

Multiply $2$ by $- 1$ .

$(1 - x) (- 2 + 2 (- x) - x \cdot - 1 - x (- x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.1.2

Multiply $- 1$ by $2$ .

$(1 - x) (- 2 - 2 x - x \cdot - 1 - x (- x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.1.3

Multiply $- x \cdot - 1$ .

Tap for more steps...

Step 3.2.1.2.1.3.1

Multiply $- 1$ by $- 1$ .

$(1 - x) (- 2 - 2 x + 1 x - x (- x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.1.3.2

Multiply $x$ by $1$ .

$(1 - x) (- 2 - 2 x + x - x (- x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$(1 - x) (- 2 - 2 x + x - 1 \cdot - 1 x \cdot x - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.2.1.2.1.5.1

Move $x$ .

$(1 - x) (- 2 - 2 x + x - 1 \cdot - 1 (x \cdot x) - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.1.5.2

Multiply $x$ by $x$ .

$(1 - x) (- 2 - 2 x + x - 1 \cdot - 1 x^{2} - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$(1 - x) (- 2 - 2 x + x + 1 x^{2} - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.1.7

Multiply $x^{2}$ by $1$ .

$(1 - x) (- 2 - 2 x + x + x^{2} - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.2.2

Add $- 2 x$ and $x$ .

$(1 - x) (- 2 - x + x^{2} - 1 \cdot 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.1.3

Multiply $- 1$ by $1$ .

$(1 - x) (- 2 - x + x^{2} - 1) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 3.2.2

Subtract $1$ from $- 2$ .

$(1 - x) (- x + x^{2} - 3) + 1 | \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} | + 0$

Step 4

Evaluate $| \begin{array}{c} 1 & - 2 \\ 1 & - 1 - x \end{array} |$ .

Tap for more steps...

Step 4.1

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

$(1 - x) (- x + x^{2} - 3) + 1 (1 (- 1 - x) - 1 \cdot - 2) + 0$

Step 4.2

Simplify the determinant.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply $- 1 - x$ by $1$ .

$(1 - x) (- x + x^{2} - 3) + 1 (- 1 - x - 1 \cdot - 2) + 0$

Step 4.2.1.2

Multiply $- 1$ by $- 2$ .

$(1 - x) (- x + x^{2} - 3) + 1 (- 1 - x + 2) + 0$

Step 4.2.2

Add $- 1$ and $2$ .

$(1 - x) (- x + x^{2} - 3) + 1 (- x + 1) + 0$

Step 5

Simplify the determinant.

Tap for more steps...

Step 5.1

Add $(1 - x) (- x + x^{2} - 3) + 1 (- x + 1)$ and $0$ .

$(1 - x) (- x + x^{2} - 3) + 1 (- x + 1)$

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

Expand $(1 - x) (- x + x^{2} - 3)$ by multiplying each term in the first expression by each term in the second expression.

$1 (- x) + 1 x^{2} + 1 \cdot - 3 - x (- x) - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

Multiply $- x$ by $1$ .

$- x + 1 x^{2} + 1 \cdot - 3 - x (- x) - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.2

Multiply $x^{2}$ by $1$ .

$- x + x^{2} + 1 \cdot - 3 - x (- x) - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.3

Multiply $- 3$ by $1$ .

$- x + x^{2} - 3 - x (- x) - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.4

Rewrite using the commutative property of multiplication.

$- x + x^{2} - 3 - 1 \cdot - 1 x \cdot x - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 5.2.2.5.1

Move $x$ .

$- x + x^{2} - 3 - 1 \cdot - 1 (x \cdot x) - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.5.2

Multiply $x$ by $x$ .

$- x + x^{2} - 3 - 1 \cdot - 1 x^{2} - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.6

Multiply $- 1$ by $- 1$ .

$- x + x^{2} - 3 + 1 x^{2} - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.7

Multiply $x^{2}$ by $1$ .

$- x + x^{2} - 3 + x^{2} - x \cdot x^{2} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.8

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

Tap for more steps...

Step 5.2.2.8.1

Move $x^{2}$ .

$- x + x^{2} - 3 + x^{2} - (x^{2} x) - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.8.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 5.2.2.8.2.1

Raise $x$ to the power of $1$ .

$- x + x^{2} - 3 + x^{2} - (x^{2} x^{1}) - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.8.2.2

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

$- x + x^{2} - 3 + x^{2} - x^{2 + 1} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.8.3

Add $2$ and $1$ .

$- x + x^{2} - 3 + x^{2} - x^{3} - x \cdot - 3 + 1 (- x + 1)$

Step 5.2.2.9

Multiply $- 3$ by $- 1$ .

$- x + x^{2} - 3 + x^{2} - x^{3} + 3 x + 1 (- x + 1)$

Step 5.2.3

Add $- x$ and $3 x$ .

$x^{2} - 3 + x^{2} - x^{3} + 2 x + 1 (- x + 1)$

Step 5.2.4

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} - 3 - x^{3} + 2 x + 1 (- x + 1)$

Step 5.2.5

Multiply $- x + 1$ by $1$ .

$2 x^{2} - 3 - x^{3} + 2 x - x + 1$

Step 5.3

Add $- 3$ and $1$ .

$2 x^{2} - x^{3} + 2 x - x - 2$

Step 5.4

Subtract $x$ from $2 x$ .

$2 x^{2} - x^{3} + x - 2$