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

• Feb 4th 2013, 05:39 PM
poorbuttryagin
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 !

Thanks.
• Feb 4th 2013, 08:02 PM
johng
Re: if I-AB is invertible , is I-BA invertible ?
Let \$\displaystyle X=(I-AB)^{-1}\$. Then \$\displaystyle I=X(I-AB)=X-XAB\$ or \$\displaystyle XAB=X-I\$. Now use this fact and the distributive law to show

\$\displaystyle (I+BXA)(I-BA)=I\$. (You know this proves the statement, don't you?)
• Feb 4th 2013, 08:11 PM
jakncoke
Re: if I-AB is invertible , is I-BA invertible ?
well assume \$\displaystyle I - AB \$ is invertible

then \$\displaystyle (I-AB)*A = (A - ABA) = A(I - BA) \$ so \$\displaystyle (I-AB)*A = A*(I-BA) \$ using the determinant rule \$\displaystyle det(I-AB)A = det(I-AB)det(A) = det(I-BA)*A) = det(I-BA)*det(A)\$ since determinants are just numbers, cancelling det(A) on both sides. \$\displaystyle det(I-AB) = det(I-BA) = 1 \$ thus I-BA is invertible.
• Feb 4th 2013, 08:17 PM
johng
Re: if I-AB is invertible , is I-BA invertible ?
Hi Jakncoke,

What if det(A) = 0?. I think your argument fails.
• Feb 4th 2013, 08:24 PM
jakncoke
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.
• Feb 4th 2013, 09:01 PM
Deveno
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:

\$\displaystyle (I + B(I - AB)^{-1}A)(I - BA) = \$

\$\displaystyle [I + B(I - AB)^{-1}A] - [BA + B(I - AB)^{-1}ABA] = \$

\$\displaystyle I - BA + B[(I - AB)^{-1} - (I - AB)^{-1}AB]A = \$

\$\displaystyle I - BA + B[(I - AB)^{-1}I - (I - AB)^{-1}AB]A = \$

\$\displaystyle I - BA + B[(I - AB)^{-1}(I - AB)]A = I - BA + BA = I\$.
• Feb 5th 2013, 03:42 AM
poorbuttryagin
Re: if I-AB is invertible , is I-BA invertible ?
Yes, i see.
Thanks everyone !!