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.

Note: This can be done easily for a 2 x 2 matrix. For larger matrices, use a calculator to find the inverse.

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

● Example, continued:

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● Example #2, continued

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