973 9.8 Matrix Inverses 6. What is the matrix equation form of the following system? 6x + 3y = 9 5x - y = 4 Provide a proof for each of the following. 7. Show that I3A = A for A = £ -2 3 0 4 5 8 0 9 -6§ and I3 = £ 1 0 0 0 1 0 0 0 1§. (This result, along with that of Example 1, illustrates that the commutative property holds when one of the matrices is an identity matrix.) 8. Let A = c a c b dd and I2 = c 1 0 0 1d . Show that AI2 = I2A = A, thus proving that I2 is the identity element for matrix multiplication for 2 * 2 square matrices. Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix In.) 9. c 5 2 7 3d and c 3 -2 -7 5d 10. c 2 1 3 1d and c -1 1 3 -2d 11. c -1 3 2 -5d and c -5 -3 -2 -1d 12. c 2 3 1 2d and c 2 -3 1 2d 13. £ 0 0 1 1 0 -1 0 -2 0§ and £ 1 1 0 0 0 -1 1 0 0§ 14. £ 1 0 0 2 1 1 0 0 0§ and £ 1 0 0 -2 1 -1 0 0 1§ 15. £ 1 0 1 0 -1 0 0 0 1§ and £ 1 0 -1 0 -1 0 0 0 1§ 16. £ 1 1 1 3 4 3 3 3 4§ and £ 7 -1 -1 -3 1 0 -3 0 1§ Find the inverse, if it exists, for each matrix. See Examples 2 and 3. 17. c -1 -2 2 -1d 18. c 1 2 -1 0d 19. c -1 3 -2 4d 20. c 3 -5 -1 2d 21. c 5 -3 10 -6d 22. c -6 -3 4 2d 23. £ 1 0 2 0 -1 1 1 0 1§ 24. £ 1 0 1 0 -1 0 0 0 1§ 25. £ 2 1 1 3 4 3 3 3 4§ 26. £ -2 -3 1 2 4 0 4 5 2§ 27. £ 2 2 -3 2 6 -3 -4 0 5§ 28. £ 2 -1 0 4 -4 1 6 -3 -1§ 29. ≥ 1 2 3 1 1 -1 3 2 0 1 2 1 2 -1 -2 0¥ 30. ≥ 1 0 -2 0 -2 1 2 2 3 -1 -2 -3 0 1 4 1¥ 31. ≥ 3 2 1 2 2 0 2 -1 0 1 -1 1 -1 2 0 1¥
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