# Linear Algebra Examples

Step 1

Consider the corresponding sign chart.

Step 2

Step 2.1

Calculate the minor for element .

Step 2.1.1

The minor for is the determinant with row and column deleted.

Step 2.1.2

Evaluate the determinant.

Step 2.1.2.1

The determinant of a matrix can be found using the formula .

Step 2.1.2.2

Simplify the determinant.

Step 2.1.2.2.1

Simplify each term.

Step 2.1.2.2.1.1

Multiply by .

Step 2.1.2.2.1.2

Multiply .

Step 2.1.2.2.1.2.1

Multiply by .

Step 2.1.2.2.1.2.2

Multiply by .

Step 2.1.2.2.2

Add and .

Step 2.2

Calculate the minor for element .

Step 2.2.1

The minor for is the determinant with row and column deleted.

Step 2.2.2

Evaluate the determinant.

Step 2.2.2.1

The determinant of a matrix can be found using the formula .

Step 2.2.2.2

Simplify the determinant.

Step 2.2.2.2.1

Simplify each term.

Step 2.2.2.2.1.1

Multiply by .

Step 2.2.2.2.1.2

Multiply by .

Step 2.2.2.2.2

Subtract from .

Step 2.3

Calculate the minor for element .

Step 2.3.1

The minor for is the determinant with row and column deleted.

Step 2.3.2

Evaluate the determinant.

Step 2.3.2.1

The determinant of a matrix can be found using the formula .

Step 2.3.2.2

Simplify the determinant.

Step 2.3.2.2.1

Simplify each term.

Step 2.3.2.2.1.1

Multiply by .

Step 2.3.2.2.1.2

Multiply by .

Step 2.3.2.2.2

Subtract from .

Step 2.4

Calculate the minor for element .

Step 2.4.1

The minor for is the determinant with row and column deleted.

Step 2.4.2

Evaluate the determinant.

Step 2.4.2.1

The determinant of a matrix can be found using the formula .

Step 2.4.2.2

Simplify the determinant.

Step 2.4.2.2.1

Simplify each term.

Step 2.4.2.2.1.1

Multiply by .

Step 2.4.2.2.1.2

Multiply .

Step 2.4.2.2.1.2.1

Multiply by .

Step 2.4.2.2.1.2.2

Multiply by .

Step 2.4.2.2.2

Subtract from .

Step 2.5

Calculate the minor for element .

Step 2.5.1

The minor for is the determinant with row and column deleted.

Step 2.5.2

Evaluate the determinant.

Step 2.5.2.1

The determinant of a matrix can be found using the formula .

Step 2.5.2.2

Simplify the determinant.

Step 2.5.2.2.1

Simplify each term.

Step 2.5.2.2.1.1

Multiply by .

Step 2.5.2.2.1.2

Multiply by .

Step 2.5.2.2.2

Add and .

Step 2.6

Calculate the minor for element .

Step 2.6.1

The minor for is the determinant with row and column deleted.

Step 2.6.2

Evaluate the determinant.

Step 2.6.2.1

The determinant of a matrix can be found using the formula .

Step 2.6.2.2

Simplify the determinant.

Step 2.6.2.2.1

Simplify each term.

Step 2.6.2.2.1.1

Multiply by .

Step 2.6.2.2.1.2

Multiply by .

Step 2.6.2.2.2

Subtract from .

Step 2.7

Calculate the minor for element .

Step 2.7.1

The minor for is the determinant with row and column deleted.

Step 2.7.2

Evaluate the determinant.

Step 2.7.2.1

The determinant of a matrix can be found using the formula .

Step 2.7.2.2

Simplify the determinant.

Step 2.7.2.2.1

Simplify each term.

Step 2.7.2.2.1.1

Multiply by .

Step 2.7.2.2.1.2

Multiply by .

Step 2.7.2.2.2

Add and .

Step 2.8

Calculate the minor for element .

Step 2.8.1

The minor for is the determinant with row and column deleted.

Step 2.8.2

Evaluate the determinant.

Step 2.8.2.1

The determinant of a matrix can be found using the formula .

Step 2.8.2.2

Simplify the determinant.

Step 2.8.2.2.1

Simplify each term.

Step 2.8.2.2.1.1

Multiply by .

Step 2.8.2.2.1.2

Multiply by .

Step 2.8.2.2.2

Add and .

Step 2.9

Calculate the minor for element .

Step 2.9.1

The minor for is the determinant with row and column deleted.

Step 2.9.2

Evaluate the determinant.

Step 2.9.2.1

The determinant of a matrix can be found using the formula .

Step 2.9.2.2

Simplify the determinant.

Step 2.9.2.2.1

Simplify each term.

Step 2.9.2.2.1.1

Multiply by .

Step 2.9.2.2.1.2

Multiply by .

Step 2.9.2.2.2

Subtract from .

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.