Hi, please choose the correct answer and provide the full explanation for each of the questions. Thank you!

11. Let M = 7 8 3 with entries from Z;. Then a basis B for the mull space M- of M is which one of the following? (a) B =3 9 . ;} (b) B = : ]} (c) B - 3 . :]} (d) B = : ]} (e) B = . . ;]} 12. Let B = {(2, 1), (3, 4) } and D = {(1, 1), (2, 3) }, both of which are ordered bases for Z?. Let M = [id], be a change of basis matrix. Which one of the following is true? (a) M = 4 6 ( b) M = 4 6 (c) M = (d) M = 3 2 6 (e) M = 4 5 5 0 2 13. Consider M = 0 20 Use the Cayley Hamilton theorem to find (M + 41)-1 where 201 I is the 3 x 3 identity matrix, working over Zs. (a) 4M2 + 3M + 1 (b) M2 + 3M + 21 (c) M2 + M + I (d) 3M2 + 4M + 1 (e) M2 + 2M + 31 14. Consider the permutations a = (1 2) (3 4) (5 6) and B = (63) (5 2)(41) expressed in cycle notation. How many permutations y of {1, 2, 3, 4, 5, 6} exist with the property B = yay where we compose from left to right? (a) 3 (b) 6 (c) 48 (d) 24 (e) 8 1 2 3 15. You are given that {a, b, c} = {1, 2, 3} and, from the 8-puzzle 4 5 6 the following 7 8 5 7 a configuration has been reached by moving squares in and out of the space: 6 b 4 8 C Which one of the following statements is true? (a) If c = 3 then b = 1. (b) If c = 2 then b = 1. (c) If c = 1 then b = 2. (d) If a = 2 then c = 3. (e) If a = 1 then c = 2