Math Matrices Test Review Key

Algebra 2 Honors Matrices Test Review Name: Key Date: Sketch the system and estimate the solutions. 1. x² + y² = 4 y = x + 4 [Image of a circle and a line intersecting at no points] No Solution 2. 4x² + y² = 16 x² = 2y² [Image of an ellipse and a parabola intersecting at four points] (2, 1.5) (-2, 1.5) (0, 0) (-2, -1.5) (2, -1.5) 3. y = x² - 2x + 1 y² = 1/2x [Image of a parabola and a sideways parabola intersecting at two points] (2, 1) (2.5, 1.2) Solve the system of equations. 4. y = x² - 3 x = 3y (3, 6) (-1, -2) 5. x² - y² = 0 3x - y = 9 (3, 0) (-3, -9) 6. 4x² + y² = 16 x² + 9 = 7y (√3, 1) (-√3, 1) (√3, -1) (-√3, -1) 7. A: x + 2y + z = 5 B: 2x - y + 2z = 2 C: 5x + 3z = 7 A + B: 3x + y + 3z = 7 C - (A + B): 2x - 2y + 2z = -5 A + B: 3x + y + 3z = 7 2B + C: 5x + 4y + 2z = 4 (5x + 3z = 7) * 2 10x + 6z = 14 (5x + 4y = 6) * 2 10x + 8y = 12 10x + 6z = 14 10x + 8y = 12 5x + 3z = 7 5x + 4y = 6 2y = -1 y = -1/2 5x + 3(-1/2) = 7 5x - 3/2 = 7 5x = 17/2 x = 17/10 (17/10, -1/2, 3) 8. A: 2x + 2y + z = 3 B: 3x - y - z = 7 C: x + y - z = -4 A + B: 5x + y = 10 2B + C: 5x + 4y = 6 (5x + y = 10) * 2 10x + 2y = 20 (5x + 4y = 6) * 2 10x + 8y = 12 10x + 2y = 20 10x + 8y = 12 6y = -8 y = -4/3 5x + 4(-4/3) = 6 5x - 16/3 = 6 5x = 34/3 x = 34/15 (-2, 1, 3) Use the following matrices to perform the indicated operation, or explain why it cannot be done. A = [3 -2] B = [1 2] C = [2 0] D = [0 2] E = [1 1] F = [1 0] G = [4 -3] H = [2 0] 9. 3E - 4G [3 3] - [16 -12] [-13 15] 10. CB [2 0] [1 2] [2 -1] [4 0] [0+0 4+0] [8-1 0+0] [0 4] [8 -1] Use the following matrices to solve the matrix equation for the unknown matrix, X, or show no solution exists. A = [2 3] B = [1 2] C = [2 5] D = [4 -3] 12. A + 3X = B 3X = B - A X = 1/3(B - A) [1/3 -1] [-5/3 2/3] 13. 2(X - 2B) = A X - 2B = 1/2A X = 1/2A + 2B [2 -1] [7 16] [7/2 1] [7 16] 14. 2X - A = 3B + 2A 2X = 3B + 3A X = 3/2(B + A) [9/2 3/2] [15/2 9] [4 1/2] [7 1/2 9] 15. 2X + 5A = C 2X = C - 5A X = 1/2(C - 5A) Not Possible b/c C - 5A 16. A + 2X = D 2X = D - A X = 1/2(D - A) Not Possible b/c D - A 17. C - 4X = D - C -4X = D - C - C -4X = D - 2C X = -1/4(D - 2C) [-5/4 1] [7/4 -1/4] [-1 1/4] [-7/4 -1/4] Evaluate. Determine if an inverse will exist. Be sure that you can do the problem by hand and on the graphing calculator |5 -3| |1 -2| 15(-2) - (-1)(3) -30 + 2 = -28 Yes! 19. |3 -4| |5 -6| 3(-6) - 5(-4) -18 + 20 = 2 Yes! 20. |-5 4| |-6 0| -5(0) - (-6)(4) 0 + 24 = 24 Yes! Evaluate. Determine if an inverse will exist. Be sure that you can do the problem by hand and on the graphing calculator. 21. |3 2| |-3 -1| 0(-1) - (-3)(2) = 6 Yes! 22. |2 3| |3 4| 2(4) - 3(3) = -1 Yes! 23. |-1 4| |2 5| -1(5) - 2(4) = -13 Yes! Find the inverse of each matrix. If the matrix has no inverse, say so. 24. |2 -3| |-4 7| 2(7) - (-4)(-3) = 2 Yes! 1/2 * |7 3| |-4 2| 25. |1 0| |0 1/2| 1(1/2) - 0(0) = 1/2 Yes! 2 * |1 0| |0 2| 26. |5 10| |-2 -4| 5(-4) - (-2)(10) = 0 No Inverse Exists Solve the system of equations using matrices. 2x2 must be done by hand, 3x3 can be done on the calculator. 27. 12x - 5y = 10 5x - 2y = 17 |12 -5| |5 -2| A⁻¹ = 1/(-24 + 25) * |-2 5| |-5 12| = |-2 5| |-5 12| |-2 5| * |10| |-5 12| |17| = |6| |2| 28. 6x - 5y = 1 8x - 7y = -1 |6 -5| |8 -7| A⁻¹ = 1/(-42 + 40) * |-7 5| |8 -6| = |-7/2 5/2| |4 -3| |-7/2 5/2| * |1| |4 -3| |-1| = |10| |-5| 29. 2x + y + 5z = 1/3 2x + 2y + 2z = 1/4 x + 3z = 1/6 |2 1 5| |2 2 2| |1 0 3| = A A⁻¹B = |6 -3 -8| * |1/3| |-1 1 1| |1/4| |-2 1 3| |1/6| = |12| |12| |-5| 30. x + y + 6z = 0 3x - y + z = 5 2x + 3z = 5 |1 1 6| |3 -1 1| |2 0 3| = A A⁻¹B = |1/2 -1/2 -1| * |0| |1/2 1/2 -1/2| |5| |-1 1 1| |5| = |10| |20| |-5|