1. Using matrices to solve Systems of Equations

2. Solving Systems with Matrices
We can use matrices to solve systems that involve 2 x 2 (2 equations, 2 variables) and 3 x 3 (3 equations, 3 variables) systems. We will look at two methods:
Cramer’s Rule (uses determinants)
Matrix Equations (uses inverse matrices)

3. Cramer’s Rule - 2 x 2 Cramer’s Rule relies on determinants
Consider the system below with
variables x and y:

4. Cramer’s Rule - 2 x 2
The formulae for the values of x and y are shown below. The numbers inside the determinants are the coefficients and constants from the equations.

5. Cramer’s Rule - 3 x 3
Consider the 3 equation system below with variables x, y and z:

6. Cramer’s Rule - 3 x 3
The formulae for the values of x, y and z are shown below. Notice that all three have the same denominator.

7. Cramer’s Rule
Not all systems have a definite solution. If the determinant of the coefficient matrix is zero, a solution cannot be found using Cramer’s Rule because of division by zero.
When the solution cannot be determined, one of two conditions exists:
The planes graphed by each equation are parallel and there are no solutions.
The three planes share one line (like three pages of a book share the same spine) or represent the same plane, in which case there are infinite solutions.

8. Cramer’s Rule
Example: 3x - 2y + z = 9 Solve the system x + 2y - 2z = x + y - 4z = -2

9. Cramer’s Rule The solution is (1, -3, 0)
3x - 2y + z = x + 2y - 2z = x + y - 4z = -2
The solution is
(1, -3, 0)

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

11. Matrix Equations Step 2: Find the inverse of the coefficient matrix.
This can be done by hand for a 2 x 2
matrix; most graphing calculators can find the inverse of a larger matrix.

12. 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!

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

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

15. 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:

16. Matrix Equations