Linear Algebra Examples

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}]$

Step 1

Find the eigenvalues.

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Step 1.1

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

$p (λ) = determinant (A - λ I_{2})$

Step 1.2

The identity matrix or unit matrix of size $2$ is the $2 \times 2$ square matrix with ones on the main diagonal and zeros elsewhere.

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

Step 1.3

Substitute the known values into $p (λ) = determinant (A - λ I_{2})$ .

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Step 1.3.1

Substitute $[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}]$ for $A$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] - λ I_{2})$

Step 1.3.2

Substitute $[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$ for $I_{2}$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] - λ [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}])$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

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

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - λ \cdot 1 & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2

Simplify each element in the matrix.

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Step 1.4.1.2.1

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - λ & - λ \cdot 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.2.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - λ & 0 λ \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.2.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - λ & 0 \\ - λ \cdot 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3

Multiply $- λ \cdot 0$ .

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Step 1.4.1.2.3.1

Multiply $0$ by $- 1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 λ & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.3.2

Multiply $0$ by $λ$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \cdot 1 \end{array}])$

Step 1.4.1.2.4

Multiply $- 1$ by $1$ .

$p (λ) = determinant ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - λ & 0 \\ 0 & - λ \end{array}])$

Step 1.4.2

Add the corresponding elements.

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 + 0 \\ - 3 + 0 & 6 - λ \end{array}]$

Step 1.4.3

Simplify each element.

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Step 1.4.3.1

Add $- 2$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 \\ - 3 + 0 & 6 - λ \end{array}]$

Step 1.4.3.2

Add $- 3$ and $0$ .

$p (λ) = determinant [\begin{array}{c} 1 - λ & - 2 \\ - 3 & 6 - λ \end{array}]$

Step 1.5

Find the determinant.

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Step 1.5.1

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

$p (λ) = (1 - λ) (6 - λ) - (- 3 \cdot - 2)$

Step 1.5.2

Simplify the determinant.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

Expand $(1 - λ) (6 - λ)$ using the FOIL Method.

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Step 1.5.2.1.1.1

Apply the distributive property.

$p (λ) = 1 (6 - λ) - λ (6 - λ) - (- 3 \cdot - 2)$

Step 1.5.2.1.1.2

Apply the distributive property.

$p (λ) = 1 \cdot 6 + 1 (- λ) - λ (6 - λ) - (- 3 \cdot - 2)$

Step 1.5.2.1.1.3

Apply the distributive property.

$p (λ) = 1 \cdot 6 + 1 (- λ) - λ \cdot 6 - λ (- λ) - (- 3 \cdot - 2)$

Step 1.5.2.1.2

Simplify and combine like terms.

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Step 1.5.2.1.2.1

Simplify each term.

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Step 1.5.2.1.2.1.1

Multiply $6$ by $1$ .

$p (λ) = 6 + 1 (- λ) - λ \cdot 6 - λ (- λ) - (- 3 \cdot - 2)$

Step 1.5.2.1.2.1.2

Multiply $- λ$ by $1$ .

$p (λ) = 6 - λ - λ \cdot 6 - λ (- λ) - (- 3 \cdot - 2)$

Step 1.5.2.1.2.1.3

Multiply $6$ by $- 1$ .

$p (λ) = 6 - λ - 6 λ - λ (- λ) - (- 3 \cdot - 2)$

Step 1.5.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$p (λ) = 6 - λ - 6 λ - 1 \cdot - 1 λ \cdot λ - (- 3 \cdot - 2)$

Step 1.5.2.1.2.1.5

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

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Step 1.5.2.1.2.1.5.1

Move $λ$ .

$p (λ) = 6 - λ - 6 λ - 1 \cdot - 1 (λ \cdot λ) - (- 3 \cdot - 2)$

Step 1.5.2.1.2.1.5.2

Multiply $λ$ by $λ$ .

$p (λ) = 6 - λ - 6 λ - 1 \cdot - 1 λ^{2} - (- 3 \cdot - 2)$

Step 1.5.2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$p (λ) = 6 - λ - 6 λ + 1 λ^{2} - (- 3 \cdot - 2)$

Step 1.5.2.1.2.1.7

Multiply $λ^{2}$ by $1$ .

$p (λ) = 6 - λ - 6 λ + λ^{2} - (- 3 \cdot - 2)$

Step 1.5.2.1.2.2

Subtract $6 λ$ from $- λ$ .

$p (λ) = 6 - 7 λ + λ^{2} - (- 3 \cdot - 2)$

Step 1.5.2.1.3

Multiply $- (- 3 \cdot - 2)$ .

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Step 1.5.2.1.3.1

Multiply $- 3$ by $- 2$ .

$p (λ) = 6 - 7 λ + λ^{2} - 1 \cdot 6$

Step 1.5.2.1.3.2

Multiply $- 1$ by $6$ .

$p (λ) = 6 - 7 λ + λ^{2} - 6$

Step 1.5.2.2

Combine the opposite terms in $6 - 7 λ + λ^{2} - 6$ .

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Step 1.5.2.2.1

Subtract $6$ from $6$ .

$p (λ) = - 7 λ + λ^{2} + 0$

Step 1.5.2.2.2

Add $- 7 λ + λ^{2}$ and $0$ .

$p (λ) = - 7 λ + λ^{2}$

Step 1.5.2.3

Reorder $- 7 λ$ and $λ^{2}$ .

$p (λ) = λ^{2} - 7 λ$

Step 1.6

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

$λ^{2} - 7 λ = 0$

Step 1.7

Solve for $λ$ .

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Step 1.7.1

Factor $λ$ out of $λ^{2} - 7 λ$ .

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Step 1.7.1.1

Factor $λ$ out of $λ^{2}$ .

$λ \cdot λ - 7 λ = 0$

Step 1.7.1.2

Factor $λ$ out of $- 7 λ$ .

$λ \cdot λ + λ \cdot - 7 = 0$

Step 1.7.1.3

Factor $λ$ out of $λ \cdot λ + λ \cdot - 7$ .

$λ (λ - 7) = 0$

Step 1.7.2

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

$λ = 0$

$λ - 7 = 0$

Step 1.7.3

Set $λ$ equal to $0$ .

$λ = 0$

Step 1.7.4

Set $λ - 7$ equal to $0$ and solve for $λ$ .

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Step 1.7.4.1

Set $λ - 7$ equal to $0$ .

$λ - 7 = 0$

Step 1.7.4.2

Add $7$ to both sides of the equation.

$λ = 7$

Step 1.7.5

The final solution is all the values that make $λ (λ - 7) = 0$ true.

$λ = 0, 7$

Step 2

The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where $N$ is the null space and $I$ is the identity matrix.

$ε_{A} = N (A - λ I_{2})$

Step 3

Find the eigenvector using the eigenvalue $λ = 0$ .

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Step 3.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + 0 [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}])$

Step 3.2

Simplify.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $0$ by each element of the matrix.

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} 0 \cdot 1 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2

Simplify each element in the matrix.

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Step 3.2.1.2.1

Multiply $0$ by $1$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} 0 & 0 \cdot 0 \\ 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.2

Multiply $0$ by $0$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} 0 & 0 \\ 0 \cdot 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.3

Multiply $0$ by $0$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} 0 & 0 \\ 0 & 0 \cdot 1 \end{array}]$

Step 3.2.1.2.4

Multiply $0$ by $1$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} 0 & 0 \\ 0 & 0 \end{array}]$

Step 3.2.2

Adding any matrix to a null matrix is the matrix itself.

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Step 3.2.2.1

Add the corresponding elements.

$[\begin{array}{c} 1 + 0 & - 2 + 0 \\ - 3 + 0 & 6 + 0 \end{array}]$

Step 3.2.2.2

Simplify each element.

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Step 3.2.2.2.1

Add $1$ and $0$ .

$[\begin{array}{c} 1 & - 2 + 0 \\ - 3 + 0 & 6 + 0 \end{array}]$

Step 3.2.2.2.2

Add $- 2$ and $0$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 + 0 & 6 + 0 \end{array}]$

Step 3.2.2.2.3

Add $- 3$ and $0$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 + 0 \end{array}]$

Step 3.2.2.2.4

Add $6$ and $0$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}]$

Step 3.3

Find the null space when $λ = 0$ .

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Step 3.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} 1 & - 2 & 0 \\ - 3 & 6 & 0 \end{array}]$

Step 3.3.2

Find the reduced row echelon form.

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Step 3.3.2.1

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 3.3.2.1.1

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 0 \\ - 3 + 3 \cdot 1 & 6 + 3 \cdot - 2 & 0 + 3 \cdot 0 \end{array}]$

Step 3.3.2.1.2

Simplify $R_{2}$ .

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

Step 3.3.3

Use the result matrix to declare the final solution to the system of equations.

$x - 2 y = 0$

$0 = 0$

Step 3.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 2 y \\ y \end{array}]$

Step 3.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \end{array}] = y [\begin{array}{c} 2 \\ 1 \end{array}]$

Step 3.3.6

Write as a solution set.

${y [\begin{array}{c} 2 \\ 1 \end{array}] | y \in R}$

Step 3.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} 2 \\ 1 \end{array}]}$

Step 4

Find the eigenvector using the eigenvalue $λ = 7$ .

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Step 4.1

Substitute the known values into the formula.

$N ([\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] - 7 [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}])$

Step 4.2

Simplify.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $- 7$ by each element of the matrix.

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - 7 \cdot 1 & - 7 \cdot 0 \\ - 7 \cdot 0 & - 7 \cdot 1 \end{array}]$

Step 4.2.1.2

Simplify each element in the matrix.

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Step 4.2.1.2.1

Multiply $- 7$ by $1$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - 7 & - 7 \cdot 0 \\ - 7 \cdot 0 & - 7 \cdot 1 \end{array}]$

Step 4.2.1.2.2

Multiply $- 7$ by $0$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - 7 & 0 \\ - 7 \cdot 0 & - 7 \cdot 1 \end{array}]$

Step 4.2.1.2.3

Multiply $- 7$ by $0$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - 7 & 0 \\ 0 & - 7 \cdot 1 \end{array}]$

Step 4.2.1.2.4

Multiply $- 7$ by $1$ .

$[\begin{array}{c} 1 & - 2 \\ - 3 & 6 \end{array}] + [\begin{array}{c} - 7 & 0 \\ 0 & - 7 \end{array}]$

Step 4.2.2

Add the corresponding elements.

$[\begin{array}{c} 1 - 7 & - 2 + 0 \\ - 3 + 0 & 6 - 7 \end{array}]$

Step 4.2.3

Simplify each element.

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Step 4.2.3.1

Subtract $7$ from $1$ .

$[\begin{array}{c} - 6 & - 2 + 0 \\ - 3 + 0 & 6 - 7 \end{array}]$

Step 4.2.3.2

Add $- 2$ and $0$ .

$[\begin{array}{c} - 6 & - 2 \\ - 3 + 0 & 6 - 7 \end{array}]$

Step 4.2.3.3

Add $- 3$ and $0$ .

$[\begin{array}{c} - 6 & - 2 \\ - 3 & 6 - 7 \end{array}]$

Step 4.2.3.4

Subtract $7$ from $6$ .

$[\begin{array}{c} - 6 & - 2 \\ - 3 & - 1 \end{array}]$

Step 4.3

Find the null space when $λ = 7$ .

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Step 4.3.1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} - 6 & - 2 & 0 \\ - 3 & - 1 & 0 \end{array}]$

Step 4.3.2

Find the reduced row echelon form.

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Step 4.3.2.1

Multiply each element of $R_{1}$ by $- \frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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Step 4.3.2.1.1

Multiply each element of $R_{1}$ by $- \frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 2 & - \frac{1}{6} \cdot 0 \\ - 3 & - 1 & 0 \end{array}]$

Step 4.3.2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ - 3 & - 1 & 0 \end{array}]$

Step 4.3.2.2

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 4.3.2.2.1

Perform the row operation $R_{2} = R_{2} + 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ - 3 + 3 \cdot 1 & - 1 + 3 (\frac{1}{3}) & 0 + 3 \cdot 0 \end{array}]$

Step 4.3.2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & 0 \\ 0 & 0 & 0 \end{array}]$

Step 4.3.3

Use the result matrix to declare the final solution to the system of equations.

$x + \frac{1}{3} y = 0$

$0 = 0$

Step 4.3.4

Write a solution vector by solving in terms of the free variables in each row.

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} - \frac{y}{3} \\ y \end{array}]$

Step 4.3.5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x \\ y \end{array}] = y [\begin{array}{c} - \frac{1}{3} \\ 1 \end{array}]$

Step 4.3.6

Write as a solution set.

${y [\begin{array}{c} - \frac{1}{3} \\ 1 \end{array}] | y \in R}$

Step 4.3.7

The solution is the set of vectors created from the free variables of the system.

${[\begin{array}{c} - \frac{1}{3} \\ 1 \end{array}]}$

Step 5

The eigenspace of $A$ is the list of the vector space for each eigenvalue.

${[\begin{array}{c} 2 \\ 1 \end{array}], [\begin{array}{c} - \frac{1}{3} \\ 1 \end{array}]}$