1. EXAMPLE 1 Evaluate determinants Evaluate the determinant of the matrix. a.54 31 SOLUTION b.2 3 4 1 1 4 3 2 0 – – – –

2. EXAMPLE 2 Find the area of a triangular region Sea Lions Off 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.

3. EXAMPLE 2 Find the area of a triangular region SOLUTION The 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 0 38 41 43 1 1 1 – – 0 + – 1 2 – 1 2 + = [(43 + 0 + 0) (0 + 0 + 1558)] – = 757.5 The area of the region is about 758 square miles. 1 0 38 41 43 1 1 1 – – 0 + 1 2 – 1 0 41 43 – 0 38 – =

4. EXAMPLE 3 Use Cramer’s rule to solve this system: 3x 5y = 21 9x + 4y = 6 – – – SOLUTION STEP 1 Evaluate the determinant of the coefficient matrix. 94 35 – Use Cramer’s rule for a 2 2 system – = 45 12 = 57 – –

5. EXAMPLE 3 STEP 2 Apply Cramer’s rule because the determinant is not 0. y = 96 321 – – – 57 – = – 189 ( 18) – – = 57 – 171 = 3 – ANSWER The solution is ( 2, 3). – Use Cramer’s rule for a 2 2 system x = 64 21 5 – – – 57 – = – 30 ( 84) – – = 57 – 114 = 2 –

6. EXAMPLE 3 CHECK Check this solution in the original equations. – 21 = Use Cramer’s rule for a 2 2 system 9x + 4y = 6 – 9( 2) + 4(3) = 6 – – ? 18 + 12 = 6 – ? – – 6 = ? 3( 2) 5(3) = 21 – –– 3x 5y = 21 – – 6 15 = 21 – ? – –

7. 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).

8. 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

9. EXAMPLE 4 Solve a multi-step problem STEP 2 Evaluate the determinant of the coefficient matrix. 6 0 1 12 0 6 2 2 2 6 0 1 0 2 = (0 + 0 + 12) (0 + 24 + 24) = 36–– STEP 3 Apply Cramer’s rule because the determinant is not 0. 180 34 12 0 6 2 2 2 6 0 1 180 44 6 2 2 34 6 0 1 12 0 180 34 44 2 C =H =O = 44 36 –––

10. EXAMPLE 4 Solve a multi-step problem 576 36 – – = – – = 432 36 – – = = 12= 1= 16 ANSWER The atomic weights of carbon, hydrogen, and oxygen are 12, 1, and 16, respectively.

11. EXAMPLE 1 Find the inverse of a 2 × 2 matrix A –1 = 1 15 – 16 5 – 8 – 2 3 Find the inverse of A =. 3 8 2 5 = – 1 = – 5 8 2 – 3 5 – 8 – 2 3

12. 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 = 1 8 – 7 4 7 1 2

13. 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 B X = 0 – 2 3 – 1 4 7 1 2 – 21 3 12 – 2 = 2 – 7 – 1 4 4 7 1 2 X X 1 0 0 1 0 – 2 3 – 1 =

14. 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 –1 and A –1 A to verify that you obtain the 3 × 3 identity matrix.

15. EXAMPLE 3 Find the inverse of a 3 × 3 matrix

16. 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. xyxy 19 – 7 = `

17. EXAMPLE 4 Solve a linear system STEP 2 Find the inverse of matrix A. 4 3 – 1 2 = A –1 = 1 8 – (–3) 4 11 1 3 2 – STEP 3 Multiply the matrix of constants by A –1 on the left. X = A –1 B = 4 11 1 3 – 2 19 – 7 = 5 – 3 = xyxy

18. 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

19. 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.

20. EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Write verbal models for the situation.

21. 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 mpdmpd 15.50 37.00 72.50 =

22. 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 –1 B. A movie pass costs $7, a package of popcorn costs $1.50, and a DVD costs $20.