Linear Algebra Examples

$[\begin{array}{c} a - b & b + c \\ - 3 d + c & 2 a - 4 d \end{array}]$

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.

$| \begin{array}{c} 2 a - 4 d \end{array} |$

Step 2.1.2

The determinant of a $1 \times 1$ matrix is the element itself.

$a_{11} = 2 a - 4 d$

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.

$| \begin{array}{c} - 3 d + c \end{array} |$

Step 2.2.2

The determinant of a $1 \times 1$ matrix is the element itself.

$a_{12} = - 3 d + c$

Step 2.3

Calculate the minor for element $a_{21}$ .

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Step 2.3.1

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

$| \begin{array}{c} b + c \end{array} |$

Step 2.3.2

The determinant of a $1 \times 1$ matrix is the element itself.

$a_{21} = b + c$

Step 2.4

Calculate the minor for element $a_{22}$ .

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Step 2.4.1

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

$| \begin{array}{c} a - b \end{array} |$

Step 2.4.2

The determinant of a $1 \times 1$ matrix is the element itself.

$a_{22} = a - b$

Step 2.5

The cofactor matrix is a matrix of the minors with the sign changed for the elements in the $-$ positions on the sign chart.

$[\begin{array}{c} 2 a - 4 d & - (- 3 d + c) \\ - (b + c) & a - b \end{array}]$