# Linear Algebra Examples

,

Step 1

Find the from the system of equations.

Step 2

Step 2.1

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

Step 2.2

Find the determinant.

Step 2.2.1

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

Step 2.2.2

Simplify the determinant.

Step 2.2.2.1

Simplify each term.

Step 2.2.2.1.1

Multiply by .

Step 2.2.2.1.2

Multiply by .

Step 2.2.2.2

Add and .

Step 2.3

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

Step 2.4

Substitute the known values into the formula for the inverse.

Step 2.5

Multiply by each element of the matrix.

Step 2.6

Simplify each element in the matrix.

Step 2.6.1

Cancel the common factor of .

Step 2.6.1.1

Cancel the common factor.

Step 2.6.1.2

Rewrite the expression.

Step 2.6.2

Combine and .

Step 2.6.3

Move the negative in front of the fraction.

Step 2.6.4

Multiply by .

Step 2.6.5

Multiply by .

Step 3

Left multiply both sides of the matrix equation by the inverse matrix.

Step 4

Any matrix multiplied by its inverse is equal to all the time. .

Step 5

Step 5.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is and the second matrix is .

Step 5.2

Multiply each row in the first matrix by each column in the second matrix.

Step 5.3

Simplify each element of the matrix by multiplying out all the expressions.

Step 6

Simplify the left and right side.

Step 7

Find the solution.