Math Matrix Answer Key

Pre-Calculus Chapter 9 Review Name:________________________________ Solve the system of equations. No calculator!! 1. 4. 3𝑥 − 𝑦 = 5 2𝑥 + 𝑦 = 5 𝑥 2 + 𝑦 2 = 10 𝑥 2 + 2𝑦 2 − 7𝑦 = 0 2. 𝑥 2 + 𝑦 2 = 10 𝑦 =𝑥+2 3. 5. 𝑥 + 𝑦 + 2𝑧 = 6 2𝑥 + 5𝑧 = 12 𝑥 + 2𝑦 + 3𝑧 = 9 6. Graph the solution set of the system of inequalities. 7. 𝑥 2 + 𝑦 2 < 16 −𝑥 + 𝑦 > 1 𝑥≥0 8. 𝑦 < −2𝑥 + 3 𝑥2 + 𝑦2 ≤ 9 𝑦 = 𝑥 2 + 2𝑥 𝑦 =6+𝑥 𝑥 + 2𝑦 + 2𝑧 = 6 𝑥−𝑦 = −1 2𝑥 + 𝑦 + 3𝑧 = 7 Give the dimensions of the matrix. 9. [ 1 2 0 1 −5 ] 3 1 10. [0 0 0 8 0 1 5 −1] 0 0 0 11. [5 6 1] Use the following matrices to perform the indicated operation, or explain why it cannot be done. (Non-Calculator!) 0 3 1 4 1 2 4 𝑨 = [2 0 −1] 𝑩=[ 𝑪 = [ 2 −1] 𝑫 = [0 −1] ] −2 1 0 −2 1 2 0 2 −1 𝑬=[ ] −5 6 4 0 𝑭 = [−1 1 7 5 2 0] 0 𝑮=[ 4 −3 ] 2 0 12. A + B 13. C – D 14. 2C + 3D 15. 5E – 2G 16. DB 17. GE 18. BC 19. BF 20. CF 21. E2 22. F(2C – D) 23. BE Use the following matrices to solve the matrix equation for the unknown matrix, X, or show that no solution exists. (Non-Calculator!) 2 1 1 −2 0 1 3 5 0 −1 𝑨=[ ] 𝑩=[ ] 𝑪=[ ] 𝑫=[ ] 3 2 −2 4 −2 4 0 2 −3 1 1 24. A + 3X = B 25. 2(X – 2B) = A 26. 2(X – A) = 3B 27. 2X + C = 5A 28. A + 2X = D 29. C – 4X = D Find the determinant |𝑨| and, if possible, the inverse 𝑨−𝟏 of the matrix. Find all determinants by hand and inverses of 2X2 matrices by hand. You can use your calculator to find inverses of 3X3 matrices. 1 30. [ 2 4 ] −9 2 31. [ 1 2 ] −3 3 33. [2 4 0 1 −3 0] −2 1 2 4 34. [−1 1 0 3 4 −12 32. [ ] −2 6 0 2] 2 1 35. [2 2 2 3 4 5] 5 6 Solve the system of equations using matrices (i.e. inverses!). (Calculator, but make sure you SHOW WORK!) 36. 12𝑥 − 5𝑦 = 10 5𝑥 − 2𝑦 = 17 37. 6𝑥 − 5𝑦 = 1 8𝑥 − 7𝑦 = −1 1 2𝑥 + 𝑦 + 5𝑧 = 3 38. 𝑥 + 2𝑦 + 2𝑧 𝑥+ 3𝑧 1 = 4 1 = 6 2𝑥 + 3𝑧 = 5 39. 𝑥 + 𝑦 + 6𝑧 = 0 3𝑥 − 𝑦 + 𝑧 = 5 Solve the system using Cramer’s Rule. (You can use your calculator here, but you must show work!) 40. 2𝑥 + 7𝑦 = 13 6𝑥 + 16𝑦 = 30 2𝑥 − 𝑦 + 5𝑧 = 0 −𝑥 + 7𝑦 =9 42. 5𝑥 + 4𝑦 + 3𝑧 = −9 41. 12𝑥 − 11𝑦 = 140 7𝑥 + 9𝑦 = 20 3𝑥 + 4𝑦 − 𝑧 = 10 − 4𝑧 = 20 43. 𝑥 2𝑥 + 𝑦 + 5𝑧 = 30 Find the area of the triangle with the given vertices. (You can use your calculator here, but you must show work!) 44. (-2, -2) (-1, 3) (3, 1) 45. (-4, 1) (1, 5) (5, -2) Answers 1. (2, 1) 2. (1, 3) (-3, -1) 9. 2 x 3 10. 3 x 4   7 6 4 16.  2  1 0     2 4 8   16 12   3   13 9  22.  5 28. Impossible 3. (-3, 3) (2, 8) 11. 1 x 3 23 17.  4  32. Det(A) = 0; No inv. 4 18.   2   22  2   1 24.  3 1  1 4  7 1  4 4    5  3 5  7   1  2 3   1 3  2 1    2  1 0  A 1 41. (8, -4) 9 17  2 17 26.  2  0 4 17   1   17  37. (6, 7)  41 31. 9 8  21.   40 41   27. Impossible 3 Det(A) = -8, A 1   8  1  8 34. Det(A) = 0, No Inv. 38. ( -1/12, 1/12, 1/12) 42.   87 , 21 , 3  43.  860 ,  540 , 10   26 26 2    2  8 7-8. See online key 15.  2 1   29 30   20. Impossible 7  2 12  6 25.  2  6. (0, 1, 2) 3 18  14. 4  5   2 2  19. 30 22 2    9 1  4   30. Det(A) = -17, 36. (65, 154) 5. (1, 1, 2)  1  1 13.  2 0    4 1  3 2  3 33. Det(A)=-1, A 1   2 1  2    8  6 9  35. Det(A)=1 A 1   2 0 40. (1/5, 9/5) 12. Impossible 23. Impossible  5  29.  4  1  4. (±√6, 2) 41 41  44. 11 units2 39. (10, 20, -5) 45. 25.5 units2 1 4  1   4 