MATRICES

Using matrices to solve Systems of Equations

Solving Systems with MatricesWe 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)

Cramer’s Rule - 2 x 2 Cramer’s Rule relies on determinants Consider the system below with

variables x and y:

a1x b1y C1

a2x b2y C2

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.

x

C1 b1

C2 b2

a1 b1

a2 b2

y

a1 C1

a2 C2

a1 b1

a2 b2

Cramer’s Rule - 3 x 3 Consider the 3 equation system below with

variables x, y and z:

a1x b1y c1z C1

a2x b2y c2z C2

a3x b3y c3z C3

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.

x

C1 b1 c1

C2 b2 c2

C3 b3 c3

a1 b1 c1

a2 b2 c2

a3 b3 c3

y

a1 C1 c1

a2 C2 c2

a3 C3 c3

a1 b1 c1

a2 b2 c2

a3 b3 c3

z

a1 b1 C1

a2 b2 C2

a3 b3 C3

a1 b1 c1

a2 b2 c2

a3 b3 c3

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.

Cramer’s Rule

x

9 2 1

5 2 2

2 1 4

3 2 1

1 2 2

1 1 4

23

231 y

3 9 1

1 5 2

1 2 4

3 2 1

1 2 2

1 1 4

69

23 3

Example: 3x - 2y + z = 9 Solve the system x + 2y - 2z = -5 x + y - 4z = -2

Cramer’s Rule

3x - 2y + z = 9 x + 2y - 2z = -5 x + y - 4z = -2

z

3 2 9

1 2 5

1 1 2

3 2 1

1 2 2

1 1 4

0

230

The solution is

(1, -3, 0)

Matrix Equations

Step 1: Write the system as a matrix equation. A three equation

system is shown below.

a1x b1y c1z C1

a2x b2y c2z C2

a3x b3y c3z C3

a1 b1 c1

a2 b2 c2

a3 b3 c3

x

y

z

C1

C2

C3

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.

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.

x

y

z

a1 b1 c1

a2 b2 c2

a3 b3 c3

1

C1

C2

C3

Note: The multiplication order on the right side is very important. We cannot multiply a 3 x 1 times a 3 x 3 matrix!

Matrix Equations Example: Solve the system

3x - 2y = 9 x + 2y = -5

3 2

1 2

x

y

9

5

3 2

1 2

1

1

8

2 2

1 3

x

y

1

8

2 2

1 3

9

5

Matrix Equations

x

y

1

8

2 2

1 3

9

5

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

x

y

1

8

8

24

x

y

1

3

Matrix Equations Example #2: Solve the 3 x 3 system

3x - 2y + z = 9 x + 2y - 2z = -5 x + y - 4z = -2

3 2 1

1 2 2

1 1 4

x

y

z

9

5

2

3 2 1

1 2 2

1 1 4

1

623

723 2

23

223

1323 7

23

123

523 8

23

Using a graphing calculator:

Matrix Equations

6 7 223 23 23

13 7223 23 23

5 8123 23 23

9

5

2

x

y

z

1

3

0

x

y

z