# Solving Matrix Equations

A matrix equation is an equation in which a variable stands for a matrix.

You can solve the simpler matrix equations using matrix addition and scalar multiplication.

Examples: 1

Solve for the matrix X:

Examples: 2

Solve for the matrix X:

## Solving systems of linear equations using matrices:

Matrix equations can be used to solve systems of linear equations by using the left and right sides of the equations.

Examples: 3

Solve the system of equations using matrices:

Write the matrix on the left as the product of coefficients and variables.

First, find the inverse of the coefficient matrix.  The inverse of is

Next, multiply each side of the matrix equation by the inverse matrix.  Since matrix multiplication is not commutative, the inverse matrix should be at the left on each side of the matrix equation.

The identity matrix on the left verifies that the inverse matrix was calculated correctly.

The solution is (4, –5).