Homework - MA 2071 Tuesday 3/25/03

Assume all matrices are nxn and invertible unless otherwise noted

1. Solve for X: CA²B^-1 XG^-1 B + N = I_n + A

2. Simplify: (I_n - A) (I_n + A + A² + . . . + A^k)

3. Solve 7x + y + 3z = 6

5x + 9y + 8z = 16

2x + 5y + 13 z = 26 by inspection. Hint: put in matrix form first.

4. Find nontrivial solutions to Ax = 2x where A is

5. Find nontrivial solutions to Ax = x where A is

6. Suppose A = P^-1DP where D is

a) Find a formula for A³⁰⁰ and A³⁹⁹

b) Does A^{-1 exist}??

7. if x1 and x2 are nontrivial solutions to Ax = 0, show that

a) x1 + x2 is also

b) kx1 is for any scalar k