GUIDED PRACTICE for Example 3

5. 2 – 2 0

2 0 – 2

12 – 4 – 6A =

Use a graphing calculator to find the inverse of the matrix A. Check the result by showing that AA-1= I and A-1A = I.

EXAMPLE 4 Solve a linear system

Use an inverse matrix to solve the linear system.

2x – 3y = 19

x + 4y = – 7

Equation 1

Equation 2

SOLUTION

STEP 1 Write the linear system as a matrix equation AX = B.

coefficient matrix of matrix of matrix (A) variables (X) constants(B)2 – 3

1 4. x

y

19

– 7=

EXAMPLE 4 Solve a linear system

STEP 2 Find the inverse of matrix A.

4 3– 1 2

=A–1 = 18 – (–3)

4111

11

3112

11–

STEP 3 Multiply the matrix of constants by A–1 on the left.

X = A–1B =

4111

11

311

11– 2

19

– 7=

5

– 3=

x

y

EXAMPLE 4 Solve a linear system

The solution of the system is (5, – 3).

ANSWER

CHECK 2(5) – 3(–3) = 10 + 9 = 195 + 4(–3) = 5 – 12 = – 7

GUIDED PRACTICE for Examples 4 and 5

Use an inverse matrix to solve the linear system.

4x + y = 103x + 5y = – 1

8.

SOLUTION

STEP 1 Write the linear system as a matrix equation AX = B.

coefficient matrix of matrix of matrix (A) variables (X) constants(B)4 1

3 5. x

y

10

– 1=

GUIDED PRACTICE for Examples 4 and 5

STEP 2 Find the inverse of matrix A.

STEP 3 Multiply the matrix of constants by A–1 on the left.

5 – 1– 3 4

=A–1 = 120 –3

5173

17

1174

17–

–

3

–2=X = A–1B =

5173

17

117

17– 4

10

– 1=

x

y

–

GUIDED PRACTICE for Examples 4 and 5

CHECK4(3) + (–2) = 10 = 12 – 2 = 10

= 10 =103x + 5y = –1

3(3) + 5(– y) = –1

9 – 10 = –1

– 1 = –1

The solution of the system is (3, –2).

ANSWER

GUIDED PRACTICE for Examples 4 and 5

9. 2x – y = – 66x – 3y = – 18

SOLUTION

STEP 1 Write the linear system as a matrix equation AX = B.

coefficient matrix of matrix of matrix (A) variables (X) constants(B)2 –1

6 –3. – x

– y

– 6

– 18=

GUIDED PRACTICE for Examples 4 and 5

STEP 2 Find the inverse of matrix A.

3 1– 6 2

=A–1 = 16 + –3

0

The inverse of matrix A is 0 so has infinitely many solutions .

ANSWER