# Proving Invertibility of AB-I & BA-I When A & B Are Both nxn Matrices

• daniel_i_l
In summary, the conversation discusses the proof of BA-I being invertible if AB-I is invertible, with the use of determinants and a proof by contradiction. The participants also suggest trying a proof by contradiction and show how to do so.
daniel_i_l
Gold Member

## Homework Statement

Q: If A and B are both nxn matrices and AB-I is invertable then prove that BA-I is also invertable.

## Homework Equations

if A is invertible iff |A|<>0

## The Attempt at a Solution

I've been thinking about this for over an hour I've only managed to prove it if either A or B are invertable. because if let's say A is invertable then:
|AB-I|<>0 => |AB-I||A|<>0 => |ABA-A|<>0 => |A||BA-I|<>0 => |BA-I|<>0 and so it's invertable. if B is invertable then you do pretty much the same thing on starting on the left side.
But what if they're both singular?
Thanks.

X is not invertible if and only if there is a v=/=0 with Xv=0.

Suppose (AB-I)v=0, and see what you can deduce. (I don't promise this works, but is the first thing that springs to mind.)

You are too hung up on determinants. Try a proof by contradiction. Assume (BA-I) is NOT invertible. Then there is a nonzero vector x such that (BA-I)x=0. Now tell me what is (AB-I)Ax=?. (By playing the same game you did with the determinants).

Thanks a lot! now i can go to sleep...

## 1. What does it mean for a matrix to be invertible?

For a matrix to be invertible, it means that there exists another matrix, called the inverse, that when multiplied with the original matrix, results in the identity matrix. This essentially means that the matrix can be "undone" or reversed.

## 2. How can you prove the invertibility of AB-I & BA-I?

One way to prove the invertibility of AB-I and BA-I is by showing that the determinant of both matrices is non-zero. If the determinant is non-zero, then the matrix is guaranteed to have an inverse. Another way is to show that the matrices have unique solutions for their respective linear systems of equations.

## 3. What is the significance of A & B being nxn matrices in this context?

The fact that A & B are both nxn matrices means that they have the same number of rows and columns. This is important because for the matrices to be invertible, they must be square matrices, meaning they have the same number of rows and columns. This also allows for the multiplication of the two matrices in both orders, AB and BA.

## 4. Can AB-I and BA-I be invertible if A & B are not invertible?

Yes, it is possible for AB-I and BA-I to be invertible even if A & B are not invertible. This is because the inverses of A & B do not necessarily need to exist for the product AB-I or BA-I to have an inverse. As long as the product results in a non-singular matrix with a non-zero determinant, it will be invertible.

## 5. Are there any other ways to prove the invertibility of AB-I & BA-I?

Yes, there are other methods that can be used to prove the invertibility of AB-I and BA-I. These include using the rank-nullity theorem, which states that for a square matrix, the rank of the matrix is equal to the number of non-zero eigenvalues. Another method is to use the diagonalization theorem, which states that a matrix is invertible if and only if it can be diagonalized.

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