Utilizing Determinants for Triangular Regions and Linear Systems

1. – – a. 5 4 b. 2 1 3 3 1 4 1 0 – – 3 4 2 EXAMPLE 1 Evaluate determinants Evaluate the determinant of the matrix. SOLUTION

2. 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. EXAMPLE 2 Find the area of a triangular region

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

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

5. Use Cramer’s rule for a 2 2 system STEP 2 Apply Cramer’s rule because the determinant is not 0. – 6 4 – – – – 5 30 ( 84) 114 21 – 2 = x = = = – – – 57 57 57 – 9 6 – – – – – 3 21 189 ( 18) 171 3 y = = = = – – – 57 57 57 ANSWER – The solution is ( 2, 3). EXAMPLE 3

6. Use Cramer’s rule for a 2 2 system – – 9x+ 4y= 6 3x5y= 21 – ? ? – – 9( 2)+ 4(3)= 6 3( 2)5(3)= 21 – – – ? ? – – – 6 15 = 21 – – 18 + 12 = 6 – 6 – 6 = – 21 – 21 = EXAMPLE 3 CHECK Check this solution in the original equations.

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. Write a linear system using the formula for each compound. Let C, H, and Orepresent the atomic weights of carbon, hydrogen, and oxygen. 6C + 12H + 6O = 180 C + 2O = 44 2H + 2O = 34 EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1

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

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

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

12. A B 2 – 7 – 1 4 – 21 3 12 – 2 X = 4 7 1 2 4 7 1 2 1 = A–1 = 8 – 7 EXAMPLE 2 Solve a matrix equation Solve the matrix equationAX = Bfor the2 × 2matrixX. SOLUTION Begin by finding the inverse of A.

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

14. 2 1 – 2 5 3 0 4 3 8 Use a graphing calculator to find the inverse of A.Then use the calculator to verify your result. A = EXAMPLE 3 Find the inverse of a 3 ×3 matrix SOLUTION Enter matrix Ainto a graphing calculator and calculate A–1. Then compute AA–1and A–1Ato verify that you obtain the 3 × 3 identity matrix.

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

16. ` STEP 1 Write the linear system as a matrix equation AX = B. x y 19 – 7 2 –3 1 4 . = EXAMPLE 4 Solve a linear system Use an inverse matrix to solve the linear system. 2x – 3y = 19 Equation 1 x + 4y = – 7 Equation 2 SOLUTION coefficient matrix of matrix of matrix (A) variables (X) constants(B)

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

18. ANSWER The solution of the system is (5, – 3). 2(5) – 3(–3) = 10 + 9 = 19 5 + 4(–3) = 5 – 12 = – 7 EXAMPLE 4 Solve a linear system CHECK

19. 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

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

21. Write a system of equations. Let mbe the cost of a movie pass, pbe the cost of a package of popcorn, and dbe the cost of a DVD. STEP 2 Rewrite the system as a matrix equation. STEP 3 2 1 0 2 2 1 4 3 2 15.50 37.00 72.50 m p d = EXAMPLE 5 Solve a multi-step problem 2m + p = 15.50 Equation 1 2m + 2p + d = 37.00 Equation 2 4m + 3p + 2d = 72.50 Equation 3

22. Enter the coefficient matrix Aand the matrix of constants Binto a graphing calculator. Then find the solution X = A–1B. STEP 4 EXAMPLE 5 Solve a multi-step problem A movie pass costs $7, a package of popcorn costs $1.50, and a DVD costs $20.