Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 6 e^{- 4 x} & 0 & - 3 12 e^{- 4 x} & 9 e^{- 2 x} & - 15 3 e^{- 4 x} & 3 e^{- 2 x} & - 3 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 6 e^{- 4 x} & 0 & - 3 \\ 12 e^{- 4 x} & 9 e^{- 2 x} & - 15 \\ 3 e^{- 4 x} & 3 e^{- 2 x} & - 3 \end{array}]$

Step 1

Consider the corresponding sign chart.

$[\begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array}]$

Step 2

Use the sign chart and the given matrix to find the cofactor of each element.

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Step 2.1

Calculate the minor for element

$a_{11}$ .

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Step 2.1.1

The minor for

$a_{11}$ is the determinant with row

$1$ and column

$1$ deleted.

$| \begin{array}{c} 9 e^{- 2 x} & - 15 \\ 3 e^{- 2 x} & - 3 \end{array} |$

Step 2.1.2

Evaluate the determinant.

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Step 2.1.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{11} = 9 e^{- 2 x} \cdot - 3 - 3 e^{- 2 x} \cdot - 15$

Step 2.1.2.2

Simplify the determinant.

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Step 2.1.2.2.1

Simplify each term.

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Step 2.1.2.2.1.1

Multiply

$- 3$ by

$9$ .

$a_{11} = - 27 e^{- 2 x} - 3 e^{- 2 x} \cdot - 15$

Step 2.1.2.2.1.2

Multiply

$- 15$ by

$- 3$ .

$a_{11} = - 27 e^{- 2 x} + 45 e^{- 2 x}$

Step 2.1.2.2.2

Add

$- 27 e^{- 2 x}$ and

$45 e^{- 2 x}$ .

$a_{11} = 18 e^{- 2 x}$

Step 2.2

Calculate the minor for element

$a_{12}$ .

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Step 2.2.1

The minor for

$a_{12}$ is the determinant with row

$1$ and column

$2$ deleted.

$| \begin{array}{c} 12 e^{- 4 x} & - 15 \\ 3 e^{- 4 x} & - 3 \end{array} |$

Step 2.2.2

Evaluate the determinant.

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Step 2.2.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{12} = 12 e^{- 4 x} \cdot - 3 - 3 e^{- 4 x} \cdot - 15$

Step 2.2.2.2

Simplify the determinant.

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Step 2.2.2.2.1

Simplify each term.

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Step 2.2.2.2.1.1

Multiply

$- 3$ by

$12$ .

$a_{12} = - 36 e^{- 4 x} - 3 e^{- 4 x} \cdot - 15$

Step 2.2.2.2.1.2

Multiply

$- 15$ by

$- 3$ .

$a_{12} = - 36 e^{- 4 x} + 45 e^{- 4 x}$

Step 2.2.2.2.2

Add

$- 36 e^{- 4 x}$ and

$45 e^{- 4 x}$ .

$a_{12} = 9 e^{- 4 x}$

Step 2.3

Calculate the minor for element

$a_{13}$ .

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Step 2.3.1

The minor for

$a_{13}$ is the determinant with row

$1$ and column

$3$ deleted.

$| \begin{array}{c} 12 e^{- 4 x} & 9 e^{- 2 x} \\ 3 e^{- 4 x} & 3 e^{- 2 x} \end{array} |$

Step 2.3.2

Evaluate the determinant.

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Step 2.3.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{13} = 12 e^{- 4 x} (3 e^{- 2 x}) - 3 e^{- 4 x} (9 e^{- 2 x})$

Step 2.3.2.2

Simplify the determinant.

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Step 2.3.2.2.1

Simplify each term.

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Step 2.3.2.2.1.1

Rewrite using the commutative property of multiplication.

$a_{13} = 12 \cdot 3 e^{- 4 x} e^{- 2 x} - 3 e^{- 4 x} (9 e^{- 2 x})$

Step 2.3.2.2.1.2

Multiply

$e^{- 4 x}$ by

$e^{- 2 x}$ by adding the exponents.

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Step 2.3.2.2.1.2.1

Move

$e^{- 2 x}$ .

$a_{13} = 12 \cdot 3 (e^{- 2 x} e^{- 4 x}) - 3 e^{- 4 x} (9 e^{- 2 x})$

Step 2.3.2.2.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a_{13} = 12 \cdot 3 e^{- 2 x - 4 x} - 3 e^{- 4 x} (9 e^{- 2 x})$

Step 2.3.2.2.1.2.3

Subtract

$4 x$ from

$- 2 x$ .

$a_{13} = 12 \cdot 3 e^{- 6 x} - 3 e^{- 4 x} (9 e^{- 2 x})$

Step 2.3.2.2.1.3

Multiply

$12$ by

$3$ .

$a_{13} = 36 e^{- 6 x} - 3 e^{- 4 x} (9 e^{- 2 x})$

Step 2.3.2.2.1.4

Rewrite using the commutative property of multiplication.

$a_{13} = 36 e^{- 6 x} - 3 \cdot 9 e^{- 4 x} e^{- 2 x}$

Step 2.3.2.2.1.5

Multiply

$e^{- 4 x}$ by

$e^{- 2 x}$ by adding the exponents.

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Step 2.3.2.2.1.5.1

Move

$e^{- 2 x}$ .

$a_{13} = 36 e^{- 6 x} - 3 \cdot 9 (e^{- 2 x} e^{- 4 x})$

Step 2.3.2.2.1.5.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a_{13} = 36 e^{- 6 x} - 3 \cdot 9 e^{- 2 x - 4 x}$

Step 2.3.2.2.1.5.3

Subtract

$4 x$ from

$- 2 x$ .

$a_{13} = 36 e^{- 6 x} - 3 \cdot 9 e^{- 6 x}$

Step 2.3.2.2.1.6

Multiply

$- 3$ by

$9$ .

$a_{13} = 36 e^{- 6 x} - 27 e^{- 6 x}$

Step 2.3.2.2.2

Subtract

$27 e^{- 6 x}$ from

$36 e^{- 6 x}$ .

$a_{13} = 9 e^{- 6 x}$

Step 2.4

Calculate the minor for element

$a_{21}$ .

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Step 2.4.1

The minor for

$a_{21}$ is the determinant with row

$2$ and column

$1$ deleted.

$| \begin{array}{c} 0 & - 3 \\ 3 e^{- 2 x} & - 3 \end{array} |$

Step 2.4.2

Evaluate the determinant.

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Step 2.4.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{21} = 0 \cdot - 3 - 3 e^{- 2 x} \cdot - 3$

Step 2.4.2.2

Simplify the determinant.

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Step 2.4.2.2.1

Simplify each term.

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Step 2.4.2.2.1.1

Multiply

$0$ by

$- 3$ .

$a_{21} = 0 - 3 e^{- 2 x} \cdot - 3$

Step 2.4.2.2.1.2

Multiply

$- 3$ by

$- 3$ .

$a_{21} = 0 + 9 e^{- 2 x}$

Step 2.4.2.2.2

Add

$0$ and

$9 e^{- 2 x}$ .

$a_{21} = 9 e^{- 2 x}$

Step 2.5

Calculate the minor for element

$a_{22}$ .

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Step 2.5.1

The minor for

$a_{22}$ is the determinant with row

$2$ and column

$2$ deleted.

$| \begin{array}{c} 6 e^{- 4 x} & - 3 \\ 3 e^{- 4 x} & - 3 \end{array} |$

Step 2.5.2

Evaluate the determinant.

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Step 2.5.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{22} = 6 e^{- 4 x} \cdot - 3 - 3 e^{- 4 x} \cdot - 3$

Step 2.5.2.2

Simplify the determinant.

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Step 2.5.2.2.1

Simplify each term.

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Step 2.5.2.2.1.1

Multiply

$- 3$ by

$6$ .

$a_{22} = - 18 e^{- 4 x} - 3 e^{- 4 x} \cdot - 3$

Step 2.5.2.2.1.2

Multiply

$- 3$ by

$- 3$ .

$a_{22} = - 18 e^{- 4 x} + 9 e^{- 4 x}$

Step 2.5.2.2.2

Add

$- 18 e^{- 4 x}$ and

$9 e^{- 4 x}$ .

$a_{22} = - 9 e^{- 4 x}$

Step 2.6

Calculate the minor for element

$a_{23}$ .

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Step 2.6.1

The minor for

$a_{23}$ is the determinant with row

$2$ and column

$3$ deleted.

$| \begin{array}{c} 6 e^{- 4 x} & 0 \\ 3 e^{- 4 x} & 3 e^{- 2 x} \end{array} |$

Step 2.6.2

Evaluate the determinant.

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Step 2.6.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{23} = 6 e^{- 4 x} (3 e^{- 2 x}) - 3 e^{- 4 x} \cdot 0$

Step 2.6.2.2

Simplify the determinant.

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Step 2.6.2.2.1

Simplify each term.

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Step 2.6.2.2.1.1

Rewrite using the commutative property of multiplication.

$a_{23} = 6 \cdot 3 e^{- 4 x} e^{- 2 x} - 3 e^{- 4 x} \cdot 0$

Step 2.6.2.2.1.2

Multiply

$e^{- 4 x}$ by

$e^{- 2 x}$ by adding the exponents.

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Step 2.6.2.2.1.2.1

Move

$e^{- 2 x}$ .

$a_{23} = 6 \cdot 3 (e^{- 2 x} e^{- 4 x}) - 3 e^{- 4 x} \cdot 0$

Step 2.6.2.2.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a_{23} = 6 \cdot 3 e^{- 2 x - 4 x} - 3 e^{- 4 x} \cdot 0$

Step 2.6.2.2.1.2.3

Subtract

$4 x$ from

$- 2 x$ .

$a_{23} = 6 \cdot 3 e^{- 6 x} - 3 e^{- 4 x} \cdot 0$

Step 2.6.2.2.1.3

Multiply

$6$ by

$3$ .

$a_{23} = 18 e^{- 6 x} - 3 e^{- 4 x} \cdot 0$

Step 2.6.2.2.1.4

Multiply

$- 3 e^{- 4 x} \cdot 0$ .

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Step 2.6.2.2.1.4.1

Multiply

$0$ by

$- 3$ .

$a_{23} = 18 e^{- 6 x} + 0 e^{- 4 x}$

Step 2.6.2.2.1.4.2

Multiply

$0$ by

$e^{- 4 x}$ .

$a_{23} = 18 e^{- 6 x} + 0$

Step 2.6.2.2.2

Add

$18 e^{- 6 x}$ and

$0$ .

$a_{23} = 18 e^{- 6 x}$

Step 2.7

Calculate the minor for element

$a_{31}$ .

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Step 2.7.1

The minor for

$a_{31}$ is the determinant with row

$3$ and column

$1$ deleted.

$| \begin{array}{c} 0 & - 3 \\ 9 e^{- 2 x} & - 15 \end{array} |$

Step 2.7.2

Evaluate the determinant.

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Step 2.7.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{31} = 0 \cdot - 15 - 9 e^{- 2 x} \cdot - 3$

Step 2.7.2.2

Simplify the determinant.

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Step 2.7.2.2.1

Simplify each term.

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Step 2.7.2.2.1.1

Multiply

$0$ by

$- 15$ .

$a_{31} = 0 - 9 e^{- 2 x} \cdot - 3$

Step 2.7.2.2.1.2

Multiply

$- 3$ by

$- 9$ .

$a_{31} = 0 + 27 e^{- 2 x}$

Step 2.7.2.2.2

Add

$0$ and

$27 e^{- 2 x}$ .

$a_{31} = 27 e^{- 2 x}$

Step 2.8

Calculate the minor for element

$a_{32}$ .

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Step 2.8.1

The minor for

$a_{32}$ is the determinant with row

$3$ and column

$2$ deleted.

$| \begin{array}{c} 6 e^{- 4 x} & - 3 \\ 12 e^{- 4 x} & - 15 \end{array} |$

Step 2.8.2

Evaluate the determinant.

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Step 2.8.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{32} = 6 e^{- 4 x} \cdot - 15 - 12 e^{- 4 x} \cdot - 3$

Step 2.8.2.2

Simplify the determinant.

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Step 2.8.2.2.1

Simplify each term.

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Step 2.8.2.2.1.1

Multiply

$- 15$ by

$6$ .

$a_{32} = - 90 e^{- 4 x} - 12 e^{- 4 x} \cdot - 3$

Step 2.8.2.2.1.2

Multiply

$- 3$ by

$- 12$ .

$a_{32} = - 90 e^{- 4 x} + 36 e^{- 4 x}$

Step 2.8.2.2.2

Add

$- 90 e^{- 4 x}$ and

$36 e^{- 4 x}$ .

$a_{32} = - 54 e^{- 4 x}$

Step 2.9

Calculate the minor for element

$a_{33}$ .

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Step 2.9.1

The minor for

$a_{33}$ is the determinant with row

$3$ and column

$3$ deleted.

$| \begin{array}{c} 6 e^{- 4 x} & 0 \\ 12 e^{- 4 x} & 9 e^{- 2 x} \end{array} |$

Step 2.9.2

Evaluate the determinant.

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Step 2.9.2.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a_{33} = 6 e^{- 4 x} (9 e^{- 2 x}) - 12 e^{- 4 x} \cdot 0$

Step 2.9.2.2

Simplify the determinant.

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Step 2.9.2.2.1

Simplify each term.

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Step 2.9.2.2.1.1

Rewrite using the commutative property of multiplication.

$a_{33} = 6 \cdot 9 e^{- 4 x} e^{- 2 x} - 12 e^{- 4 x} \cdot 0$

Step 2.9.2.2.1.2

Multiply

$e^{- 4 x}$ by

$e^{- 2 x}$ by adding the exponents.

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Step 2.9.2.2.1.2.1

Move

$e^{- 2 x}$ .

$a_{33} = 6 \cdot 9 (e^{- 2 x} e^{- 4 x}) - 12 e^{- 4 x} \cdot 0$

Step 2.9.2.2.1.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a_{33} = 6 \cdot 9 e^{- 2 x - 4 x} - 12 e^{- 4 x} \cdot 0$

Step 2.9.2.2.1.2.3

Subtract

$4 x$ from

$- 2 x$ .

$a_{33} = 6 \cdot 9 e^{- 6 x} - 12 e^{- 4 x} \cdot 0$

Step 2.9.2.2.1.3

Multiply

$6$ by

$9$ .

$a_{33} = 54 e^{- 6 x} - 12 e^{- 4 x} \cdot 0$

Step 2.9.2.2.1.4

Multiply

$- 12 e^{- 4 x} \cdot 0$ .

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Step 2.9.2.2.1.4.1

Multiply

$0$ by

$- 12$ .

$a_{33} = 54 e^{- 6 x} + 0 e^{- 4 x}$

Step 2.9.2.2.1.4.2

Multiply

$0$ by

$e^{- 4 x}$ .

$a_{33} = 54 e^{- 6 x} + 0$

Step 2.9.2.2.2

Add

$54 e^{- 6 x}$ and

$0$ .

$a_{33} = 54 e^{- 6 x}$

Step 2.10

The cofactor matrix is a matrix of the minors with the sign changed for the elements in the

$-$ positions on the sign chart.

$[\begin{array}{c} 18 e^{- 2 x} & - 9 e^{- 4 x} & 9 e^{- 6 x} \\ - 9 e^{- 2 x} & - 9 e^{- 4 x} & - 18 e^{- 6 x} \\ 27 e^{- 2 x} & 54 e^{- 4 x} & 54 e^{- 6 x} \end{array}]$