1. Matrix Equations
Step 1: Write the system as a matrix equation. A three-equation system is shown below.

2. Matrix Equations
Step 2: Find the inverse of the coefficient matrix. Note: This can be done easily for a 2 x 2 matrix. For larger matrices, use a calculator to find the inverse.

3. Matrix Equations
Step 3: Multiply both sides of the matrix equation by the inverse. The inverse of the coefficient matrix times the coefficient matrix equals the identity matrix.
Note: The multiplication order on the right side is very important. We cannot multiply a 3 x 1 times a 3 x 3 matrix!

4. Matrix Equations
Example: Solve the system 3x - 2y = x + 2y = -5

5. Matrix Equations
Example, continued:
Multiply the matrices (a ‘2 x 2’ times a ‘2 x 1’) first, then distribute the scalar.

6. Matrix Equations
Example #2: Solve the 3 x 3 system 3x - 2y + z = 9 x + 2y - 2z = -5 x + y - 4z = -2
Using a graphing calculator,

7. Matrix Equations
Example #2, continued