610 The notation Ak means the matrix A multiplied with itself k times. 6. (a) For the 2

12 Suppose A, B, and C are 2 3 2 matrices, E, F, and G are 3 3 3 matrices, H and K are 2 3 3 matrices, and L and M are 3 3 2 matrices. For each of the following, if the operation is defined, specify the size of the matrix that results. 1. (a) A 1 4C (b) 1 2K 1 L (c) 5K 1 3H (d) 0G 1 3sE 1 Fd (e) 3A 1 6sB 1 Md (f) 12M 2 L (g) F 1 G 2 2C (h) F 1 G, where  and  are scalars 2. (a) AB 1 C (b) 3GF (c) CK 1 B (d) CK 1 H (e) EMC (f) GLH (g) HLG (h) 2EL 1 5MB

12 Suppose A, B, and C are 2 3 2 matrices, E, F, and G are 3 3 3 matrices, H and K are 2 3 3 matrices, and L and M are 3 3 2 matrices. For each of the following, if the operation is defined, specify the size of the matrix that results. 1. (a) A 1 4C (b) 1 2K 1 L (c) 5K 1 3H (d) 0G 1 3sE 1 Fd (e) 3A 1 6sB 1 Md (f) 12M 2 L (g) F 1 G 2 2C (h) F 1 G, where  and  are scalars 2. (a) AB 1 C (b) 3GF (c) CK 1 B (d) CK 1 H (e) EMC (f) GLH (g) HLG (h) 2EL 1 5MB

Using the given matrices, calculate the quantities, if they are defined. A c 2 5 1 7d B c 7 x a 5 d C c 9 2 7 10d E 0 3 1 7 6 0 9 13 5 F x 2 9 6 y 13 0 1 0 G 13 2 0 5 9 12 7 0 1 H c 3 1 7 15 0 2d K c 1 0 1 0 y 1 d (a) A 2 3C (b) 3F 1 G 2 E (c) 5K 1 9H (d) H 2 12G (e) 5B 2 A (f) B 2 9K (g) 3F 2 F (h) H 2 K 1 2H

Using the given matrices, calculate the quantities, if they are defined. A c 7 0 1 9d B c 0 2 13 6d C c 15 0 0 2d E x 5 3 0 9 1 6 4 0 F y 0 y 0 1 0 0 1 1 G 1 2 3 5 4 6 9 7 8 H c 2 1 3 3 7 6d K 9 2 1 5 7 8 (a) A 2 2BC (b) EG (c) FK 2 2K (d) HEA (e) 3BC 2 HK (f) 7K 2 H (g) AHKB (h) BKHA 5. Find the transpose of each matrix. (a) 3X 1 2 (b) f3 3 9g (c) c 2 1 7 8 3 6d (d) c 2 1 1 3

610 The notation Ak means the matrix A multiplied with itself k times. 6. (a) For the 2 3 2 identity matrix I, show that I 2 I. (b) For the n 3 n identity matrix I, show that I 2 I. (c) What do you think the entries of I k are?

610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.

610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.

610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.

610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.

610 The notation Ak means the matrix A multiplied with itself k times. 7. (a) Show that, for all n 3 n diagonal matrices D having entries dii, the matrix D2 is also diagonal with entries dii 2 . (b) For an arbitrary n 3 n diagonal matrix D, what do you think the entries of Dk are? 8. Show that, in general, if A is a 2 3 2 matrix, then c a11 a12 a21 a22 d 2 c a11 2 a21 2 a12 2 a22 2 d 9. Consider the matrix C c 0 1 1 0d. (a) Calculate C 2 , C 3 , C 4 , and C 5 . (b) What do you think C k is? 10. Find an example of a nonzero 2 3 2 matrix whose square is the zero matrix.

For arbitrary 2 3 2 matrices A, B, and C, verify the following properties. (a) AsBCd sABdC (b) AsB 1 Cd AB 1 AC (c) sB 1 CdA BA 1 CA

Suppose that A and B are arbitrary 2 3 2 matrices. Is the quantity given always equal to sA 1 Bd 2 ? (a) A2 1 2AB 1 B2 (b) sB 1 Ad 2 (c) AsA 1 Bd 1 BsA 1 Bd (d) A2 1 AB 1 BA 1 B

Suppose that A is a 2 3 2 matrix that commutes with all possible 2 3 2 matrices. Show that, necessarily, a11 a22 and a12 a21 0. Hint: Try using X c 1 0 0 0d and Y c 0 1 0 0d as two particular 2 3 2 matrices with which A must commute.

a) For any 2 3 2 matrices A and B show that sA 1 Bd T AT 1 BT (b) For any n 3 n matrices A and B show that sA 1 Bd T AT 1 BT

(a) For any 2 3 2 matrices A and B show that sABd T BT AT (b) For any n 3 n matrices A and B show that sABd T BT AT

A matrix A is called symmetric if AT A. Verify, for all 3 3 3 matrices B, that B 1 BT is symmetric

Suppose a matrix A satisfies the equation A 1 AT 0. What must be true about the entries of A? Such matrices are called skew-symmetric

An upper triangular matrix is a square matrix with all zeros below its diagonal. A lower triangular matrix is a square matrix with all zeros above its diagonal. (a) For all 3 3 3 upper triangular matrices U, verify that U 2 is also upper triangular. (b) For all 3 3 3 lower triangular matrices L, verify that L2 is also lower triangular.