# Homework Help: Linear Algebra- prove that A is similar to B then A inverse is similar to B invese.

1. Feb 18, 2010

### zeion

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

If A and B are invertible matrices and B is similar to A, prove that B-1 is similar to A-1

2. Relevant equations

3. The attempt at a solution

Not sure how to do this.. I know that similar matrices have the same characteristic polynomials and the same eigenvalues and same determinant.. but I'm not sure how to tie that in with their inverses..

2. Feb 18, 2010

### Dick

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

What's the definition of 'similar'?

3. Feb 18, 2010

### Staff: Mentor

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

There are some definitions and equations that you should have included amongst your relevant equations - namely, the definition of similarity and the formula for the inverse of a product of invertible matrices.

4. Feb 18, 2010

### zeion

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

If A and B are similar there exists an invertible matrix P such that B = P-1AP.

5. Feb 18, 2010

### Fredrik

Staff Emeritus
Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

Suppose X and Y are n×n matrices. Is there something that you can multiply XY with (either from the left or from the right) to get the identity matrix? When you have answered that, you have an explcity formula for (XY)-1, which you can use to rewrite the expression for B-1 that you already have.

6. Feb 18, 2010

### Staff: Mentor

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

How about the formula for the inverse of a product of matrices, where both matrices have inverses?

7. Feb 18, 2010

### zeion

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

(xy)(xy)-1 = (xy)-1(xy) = i?

8. Feb 18, 2010

### Staff: Mentor

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

No, what I'm asking about is: Do you know a formula for (AB)-1?

You should try to get into the habit of using caps for matrices. Using i for the identity matrix could easily be interpreted as the imaginary unit i.

9. Feb 18, 2010

### zeion

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

( AB )-1 = ( B -1 A -1)?

My letters keep getting lower cased on their own??

10. Feb 18, 2010

### zeion

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

So then I get

(AB)-1 = B-1A-1
(A(P-1AP))-1 = B-1A-1
(A(P-1AP))-1A = B-1A-1A
B-1 = (A(P-1AP))-1A

..?

11. Feb 18, 2010

### vela

Staff Emeritus
Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

Start with the definition of similarity, $B=P^{-1}AP$, and invert both sides using the formula for the inverse of a product of matrices.

12. Feb 18, 2010

### Fredrik

Staff Emeritus
Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

In other words, figure out how to generalize $(XY)^{-1}$ to three matrices (What is $(XYZ)^{-1}$?), and use it to express $B^{-1}$ in a more useful way.

13. Feb 18, 2010

### zeion

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

Oh this is kind of brilliant
B = P-1AP
(B)-1 = (P-1AP)-1
B-1 = P-1A-1(P-1)-1
B-1 = P-1A-1P

Oh man how are you guys so smart

14. Feb 18, 2010

### zeion

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

For this next question it asks me to prove that if B is similar to A, then BT is similar to AT.. so I try the same thing but I get

BT = PTAT(PT)-1?

15. Feb 18, 2010

### VeeEight

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

Looks right to me. If you want it in the form BT=Q-1AQ, then take Q=(PT)-1

16. Feb 18, 2010

### zeion

Re: Linear Algebra- prove that A is similar to B then A inverse is similar to B inves

Oh ok great that makes sense