Exercise 4.4

Let A = 3 7 2 5     and B = 6 8 7 9     . Verify that (AB) -1 = B -1 A-1.

If A = 3 1 1 2−    , show that A2 - 5A + 7I = O. Hence find A-1.

For the matrix A = 3 2 1 1    , find the numbers a and b such that A2 + a A + bI = O.

For the matrix A = 1 1 1 1 2 3 2 1 3 − −           Show that A3- 6A2 + 5A + 11 I = O. Hence, find A-1.

If A = 2 1 1 1 2 1 1 1 2 − − − −           Verify that A3 - 6A2 + 9A - 4I = O and hence find A-1

Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to (A) |A | (B) |A |2 (C) |A |3 (D) 3|A |

If A is an invertible matrix of order 2, then det (A-1) is equal to (A) det (A) (B) det (A) (C) 1 (D) 0