if I-AB is invertible , is I-BA invertible ?

Hi, everyone ~

I read Linear Algebra by Hoffman & Kunze.

At 190pg #8,

A, B := n*n matrices . Prove that if I-AB is invertible, then I-BA is invertible and (I-BA)^(-1) = I +B(I-AB)^(-1) A.

Any hint or comment are welcomed !

Please help !

Thanks.

Re: if I-AB is invertible , is I-BA invertible ?

Let . Then or . Now use this fact and the distributive law to show

. (You know this proves the statement, don't you?)

Re: if I-AB is invertible , is I-BA invertible ?

well assume is invertible

then so using the determinant rule since determinants are just numbers, cancelling det(A) on both sides. thus I-BA is invertible.

Re: if I-AB is invertible , is I-BA invertible ?

Hi Jakncoke,

What if det(A) = 0?. I think your argument fails.

Re: if I-AB is invertible , is I-BA invertible ?

Quote:

Originally Posted by

**johng** Hi Jakncoke,

What if det(A) = 0?. I think your argument fails.

yes you are right! what was i thinking.

Re: if I-AB is invertible , is I-BA invertible ?

since they actually give the answer, i don't see what the fuss is about:

.

Re: if I-AB is invertible , is I-BA invertible ?

Yes, i see.

Thanks everyone !!