LINEAR ALGEBRA (Math 211)
HW #7 (Due Wed. Oct. 18 )

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

5.

(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 .