EXAMPLE 1 Evaluate determinants

Evaluate the determinant of the matrix.

a. 5 43 1

SOLUTION

b. 2

34

114

3

20

– –

– –

EXAMPLE 2 Find the area of a triangular region

Sea LionsOff the coast of California lies a triangular region of the Pacific Ocean where huge populations of sea lions and seals live. The triangle is formed by imaginary lines connecting Bodega Bay, the Farallon Islands, and Año Nuevo Island, as shown. (In the map, the coordinates are measured in miles.) Use a determinant to estimate the area of the region.

EXAMPLE 2 Find the area of a triangular region

SOLUTIONThe approximate coordinates of the vertices of the triangular region are ( 1, 41), (38, 43), and (0, 0). So, the area of the region is:

– –

Area =1

038

4143

1

11

– –

0+ –

12

–12

+= [(43 + 0 + 0) (0 + 0 + 1558)] –

= 757.5

The area of the region is about 758 square miles.

1

038

4143

1

11

– –

0+ 1

2 –1

0

4143 –0

38 –

=

EXAMPLE 3

Use Cramer’s rule to solve this system:3x 5y = 219x + 4y = 6

– – –

SOLUTION

STEP 1Evaluate the determinant of the coefficient matrix.

9 43 5 –

Use Cramer’s rule for a 2 2 system

–= 45 12 = 57 – –

EXAMPLE 3

STEP 2Apply Cramer’s rule because the determinant is not 0.

y =

9 63 21 – –

–

57 – = 57 – 189 ( 18) – –

= 57 –171

= 3 –

ANSWER

The solution is ( 2, 3). –

Use Cramer’s rule for a 2 2 system

x =

6 421 5 – –

–

57 – = 57 – 30 ( 84) – –

= 57 –114

= 2 –

EXAMPLE 3

CHECK

Check this solution in the original equations.

– 21= – 21

Use Cramer’s rule for a 2 2 system

9x + 4y = 6 –9( 2) + 4(3) = 6 – – ?

18 + 12 = 6 – ? – – 6= – 6

?3( 2) 5(3) = 21 – ––3x 5y = 21 ––

6 15 = 21 – ? – –

EXAMPLE 4 Solve a multi-step problem

CHEMISTRY

The atomic weights of three compounds are shown. Use a linear system and Cramer’s rule to find the atomic weights of carbon (C), hydrogen (H), and oxygen (O).

EXAMPLE 4 Solve a multi-step problem

SOLUTION

Write a linear system using the formula for each compound. Let C, H, and O represent the atomic weights of carbon, hydrogen, and oxygen.

6C + 12H + 6O = 180 C + 2O = 44

2H + 2O = 34

STEP 1

EXAMPLE 4 Solve a multi-step problem

STEP 2Evaluate the determinant of the coefficient matrix.

6

01

120

6

22

2

6

01

1202

= (0 + 0 + 12) (0 + 24 + 24) = 36– –

STEP 3Apply Cramer’s rule because the determinant is not 0.

180

34

120

6

22

2

6

01

180 44

6

22

34

6

01

120

180

3444

2C = H = O =

44

36 36 36– – –

EXAMPLE 4 Solve a multi-step problem

576 36–

–=

3636–

–=

43236–

–=

= 12 = 1 = 16

ANSWER

The atomic weights of carbon, hydrogen, and oxygen are 12, 1, and 16, respectively.

EXAMPLE 1 Find the inverse of a 2 × 2 matrix

A–1 = 115 – 16

5 – 8

– 2 3

Find the inverse of A = .

3 8

2 5

= – 1 =– 5 8

2 – 3

5 – 8

– 2 3

EXAMPLE 2 Solve a matrix equation

SOLUTION

Begin by finding the inverse of A.

4 7

1 2=

Solve the matrix equation AX = B for the 2 × 2 matrix X.

2 – 7

– 1 4

– 21 3

12 – 2

A B

X =

A–1 = 18 – 7

4 7

1 2

EXAMPLE 2 Solve a matrix equation

To solve the equation for X, multiply both sides of the equation by A– 1 on the left.

A–1 AX = A–1 B

IX = A–1 B

X = A–1 BX =0 – 2

3 – 1

4 7

1 2

– 21 3

12 – 2=

2 – 7

– 1 4

4 7

1 2X

X 1 0

0 1

0 – 2

3 – 1=

EXAMPLE 3 Find the inverse of a 3 × 3 matrix

Use a graphing calculator to find the inverse of A.Then use the calculator to verify your result.

2 1 – 2

5 3 0

4 3 8

A =

SOLUTION

Enter matrix A into a graphing calculator and calculate A–1. Then compute AA–1and A–1A to verify that you obtain the 3 × 3 identity matrix.

EXAMPLE 3 Find the inverse of a 3 × 3 matrix

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 = 19

5 + 4(–3) = 5 – 12 = – 7

EXAMPLE 5 Solve a multi-step problem

Gifts

A company sells three types of movie gift baskets. A basic basket with 2 movie passes and 1 package of microwave popcorn costs $15.50. A medium basket with 2 movie passes, 2 packages of popcorn, and 1 DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $72.50. Find the cost of each item in the gift baskets.

EXAMPLE 5 Solve a multi-step problem

SOLUTION

STEP 1 Write verbal models for the situation.

EXAMPLE 5 Solve a multi-step problem

STEP 2 Write a system of equations. Let m be the cost of a movie pass, p be the cost of a package of popcorn, and d be the cost of a DVD.

2m + p = 15.50 Equation 1

2m + 2p + d = 37.00 Equation 2

4m + 3p + 2d = 72.50 Equation 3

STEP 3 Rewrite the system as a matrix equation.

2 1 0

2 2 1

4 3 2

m

p

d

15.50

37.00

72.50

=

EXAMPLE 5 Solve a multi-step problem

STEP 4 Enter the coefficient matrix A and the matrix of constants B into a graphing calculator. Then find the solution X = A–1B.

A movie pass costs $7, a package of popcorn costs $1.50, and a DVD costs $20.