Proving Similarity of Inverse Matrices

In summary: So if you have two matrices A and B with inverse matrices P and Q, then PQ=I?If A and B are similar there exists an invertible matrix P such that B = P-1AP.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.How about the formula for the inverse of a product of matrices, where both matrices have inverses?suppose x and y
  • #1
zeion
466
1

Homework Statement



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

Homework Equations





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..
 
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  • #2


What's the definition of 'similar'?
 
  • #3


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


If A and B are similar there exists an invertible matrix P such that B = P-1AP.
 
  • #5


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


How about the formula for the inverse of a product of matrices, where both matrices have inverses?
 
  • #7


fredrik said:
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.
mark44 said:
how about the formula for the inverse of a product of matrices, where both matrices have inverses?

(xy)(xy)-1 = (xy)-1(xy) = i?
 
  • #8


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


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

My letters keep getting lower cased on their own??
 
  • #10


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


Start with the definition of similarity, [itex]B=P^{-1}AP[/itex], and invert both sides using the formula for the inverse of a product of matrices.
 
  • #12


In other words, figure out how to generalize [itex](XY)^{-1}[/itex] to three matrices (What is [itex](XYZ)^{-1}[/itex]?), and use it to express [itex]B^{-1}[/itex] in a more useful way.
 
  • #13


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


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


Looks right to me. If you want it in the form BT=Q-1AQ, then take Q=(PT)-1
 
  • #16


Oh ok great that makes sense
 

1. What is the definition of inverse matrices?

Inverse matrices are two square matrices that, when multiplied together, result in the identity matrix.

2. How do you prove the similarity of inverse matrices?

To prove the similarity of inverse matrices, you need to show that the two matrices have the same eigenvalues and eigenvectors. This can be done by calculating the determinants and eigenvectors of both matrices and comparing them.

3. Can two matrices be similar but not have inverse matrices?

No, two matrices can only have inverse matrices if they are similar. If two matrices are not similar, they will have different eigenvalues and eigenvectors, and thus cannot have inverse matrices.

4. What is the significance of proving similarity of inverse matrices?

Proving the similarity of inverse matrices is important because it allows us to find the inverse of a matrix without having to perform time-consuming calculations. It also helps in solving systems of linear equations and other mathematical problems.

5. Are all square matrices similar to their inverse matrices?

No, not all square matrices are similar to their inverse matrices. Only matrices that have the same eigenvalues and eigenvectors can be similar to their inverse matrices.

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