1. For matrix compute .
2. Let A and B be 3 x 3 matrices with and . Find the value of
3. Let E1, E2, E3 be 3 x 3 elementary matrices of types (1), (2), and (3), respectively, and let A be a 3 x 3 matrices with . Assume, additionally, that E1 corresponds to multiplication of the second row by 3. Find the values of each of the following
4. Assuming that compute the following determinants
(a) A matrix A is said to be idempotent if A2=A. Prove that if A is idempotent, then or 1.
(b) A matrix A is said to be nilpotent if Ak=0 for some positive integer k. Prove that if A is nilpotent, then .
(c) A matrix A is said to be orthogonal if . Prove that if A is orthogonal, then or -1.
(d) An n x n matrix A is said to be skew symmetric if AT=-A. Prove that if A is sew symmetric and n is odd, then .