Matrix Algebra Problems II

due Tuesday

Ma 2071

Assume that all matrices are square, nxn, unless otherwise noted.

1. show directly that (AB)^{-1 =} B^-1A^-1

2. What is (A^-1)^{-1 ?}

3. Find (A^-1 CB)^-1 (note correction on exponent)

4. If A = P D P^-1 what is A² ? A³ ? A^k?

5. Expand (A + B)²

6. Expand (I_n + A)³

7. Expand and simplify (I_n - A)(I_n + A + A² + A³)

8. If D is a 3x3 "diagonal" matrix with scalars d₁, d₂ and d₃ on it's diagonal and 0's elsewhere, find D² and D³. Generalize to D^k

9. In problem 8, if it is also given that -1 < d_i < +1 what is the limit as k ->∞ of D^k ?

10. Suppose that A = P D P^-1 and that it is known that D is a diagonal matrix with either +1 or -1 on its diagonal and 0s elsewhere. What is A²?

A⁴? A⁶? How about A³? A⁵ ?

11. Solve for X in the matrix equation CB^-1XA + D = F - I_n

12. If A = P D P^-1 , solve for D. What is D^{-1 ?}