A given by 1. M₁1 4. M41 12. A = 14. A = 16. A = A = 2-1 3 1 4 13-1 624 1 2 20-2 2. M₂1 5. M34 12 4 23 7 4 2 10 1 2 1 032 -1 1 1 200 3 1 0 24 2 3. M31 6. M43 13. A = 15. A = 17. A = (7) 2-3 2 1 -1 -2 3 200 132 214 1215 0300 0412 0314 In Exercises 8-19, calculate the determinant of the given matrix. Use Theorem 3 to state whether the matrix is singular or nonsingular. 31] -1 2 2 3 46 8. A = 10. A = 9. A = 11. A = 1 -1 -2 21 2 1 1 21 4.2 Determinants and the Eigenvalue Problem 289 23. Let A = (aij) be the (n × n) matrix specified thus: aij = d for i = j and ajj = 1 for i # j. For n = 2, 3, and 4, show that det (A) = (d 1)"-¹(d - 1 + n). 24. Let A and B be (n x n) matrices. Use Theorems 2 and 3 to give a quick proof of each of the following. a) If either A or B is singular, then AB is singular. b) If AB is singular, then either A or B is singular. 25. Suppose that A is an (n × n) nonsingular matrix, and recall that det (I) = 1, where I is the (n x n) identity matrix. Show that det (A-¹) = 1/det(A). 26. If A and B are (nxn) matrices, then usually AB # BA. Nonetheless, argue that always det(AB) =