# A^k matrix singularity and (A^k)^-1 = (A^-1)^k

1. Jul 13, 2009

### ramtin

1. The problem statement, all variables and given/known data

Let A be nonsingular. Prove That for any positive integer k , A^k is nonsingular, And (A^k)^-1 = (A^-1)^k.

2. Relevant equations

3. The attempt at a solution

Last edited: Jul 13, 2009
2. Jul 13, 2009

### cipher42

What have you tried so far?

3. Jul 13, 2009

Fill this in: a problem that requires you to prove a simple expression is true for every positive integer $$k$$ is a good candidate for m ************ ******n

4. Jul 13, 2009

### ramtin

You answer was not complete ...What are the * ?
Please somebody help me!

5. Jul 13, 2009

### Office_Shredder

Staff Emeritus
Start small. Can you prove it's true for k=2? How can you generalize the proof?

6. Jul 13, 2009

### ramtin

I can't prove it for 2 ,,,don't know How to generalize that

7. Jul 13, 2009

### Office_Shredder

Staff Emeritus
Start by contradiction. If A is nonsingular, then if A2 is singular what can we prove about A? Try messing around with the equation A2v = 0

8. Jul 13, 2009

### HallsofIvy

Then what do you know? Under what conditions is a matrix "singular"? What does having an inverse mean?