# Linear Algebra Examples

Step 1

Step 1.1

Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.

Step 1.1.1

Consider the corresponding sign chart.

Step 1.1.2

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

Step 1.1.3

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

Step 1.1.4

Multiply element by its cofactor.

Step 1.1.5

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

Step 1.1.6

Multiply element by its cofactor.

Step 1.1.7

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

Step 1.1.8

Multiply element by its cofactor.

Step 1.1.9

Add the terms together.

Step 1.2

Evaluate .

Step 1.2.1

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

Step 1.2.2

Simplify the determinant.

Step 1.2.2.1

Simplify each term.

Step 1.2.2.1.1

Multiply by .

Step 1.2.2.1.2

Multiply by .

Step 1.2.2.2

Subtract from .

Step 1.3

Evaluate .

Step 1.3.1

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

Step 1.3.2

Simplify the determinant.

Step 1.3.2.1

Simplify each term.

Step 1.3.2.1.1

Multiply by .

Step 1.3.2.1.2

Multiply by .

Step 1.3.2.2

Subtract from .

Step 1.4

Evaluate .

Step 1.4.1

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

Step 1.4.2

Simplify the determinant.

Step 1.4.2.1

Simplify each term.

Step 1.4.2.1.1

Multiply by .

Step 1.4.2.1.2

Multiply by .

Step 1.4.2.2

Subtract from .

Step 1.5

Simplify the determinant.

Step 1.5.1

Simplify each term.

Step 1.5.1.1

Multiply by .

Step 1.5.1.2

Multiply by .

Step 1.5.1.3

Multiply by .

Step 1.5.2

Add and .

Step 1.5.3

Subtract from .

Step 2

Since the determinant is non-zero, the inverse exists.

Step 3

Set up a matrix where the left half is the original matrix and the right half is its identity matrix.

Step 4

Step 4.1

Multiply each element of by to make the entry at a .

Step 4.1.1

Multiply each element of by to make the entry at a .

Step 4.1.2

Simplify .

Step 4.2

Perform the row operation to make the entry at a .

Step 4.2.1

Perform the row operation to make the entry at a .

Step 4.2.2

Simplify .

Step 4.3

Perform the row operation to make the entry at a .

Step 4.3.1

Perform the row operation to make the entry at a .

Step 4.3.2

Simplify .

Step 4.4

Multiply each element of by to make the entry at a .

Step 4.4.1

Multiply each element of by to make the entry at a .

Step 4.4.2

Simplify .

Step 4.5

Perform the row operation to make the entry at a .

Step 4.5.1

Perform the row operation to make the entry at a .

Step 4.5.2

Simplify .

Step 4.6

Multiply each element of by to make the entry at a .

Step 4.6.1

Multiply each element of by to make the entry at a .

Step 4.6.2

Simplify .

Step 4.7

Perform the row operation to make the entry at a .

Step 4.7.1

Perform the row operation to make the entry at a .

Step 4.7.2

Simplify .

Step 4.8

Perform the row operation to make the entry at a .

Step 4.8.1

Perform the row operation to make the entry at a .

Step 4.8.2

Simplify .

Step 4.9

Perform the row operation to make the entry at a .

Step 4.9.1

Perform the row operation to make the entry at a .

Step 4.9.2

Simplify .

Step 5

The right half of the reduced row echelon form is the inverse.