Invertible matrices problems

[diogomgf]

Hello everyone,

I have a linear algebra problem that I need help with, and another problem I think solved but I'm not 100% sure and the book doesn't have the solution for it.
The problems are as follow:

1- \(\displaystyle A, B \) are matrices \(\displaystyle \in M_{n*n} \), with \(\displaystyle A\) being invertible. Show that \(\displaystyle A + B \) and \(\displaystyle I_n + BA^{-1} \) are either both invertible or both non-invertible.

I don't really know where to start on this problem... If some one could hint me in the right direction...

2- Let \(\displaystyle A, B \) be matrices \(\displaystyle \in M_{n*n} \) such that \(\displaystyle A, B \) and \(\displaystyle A-B \) are all invertible. Justify that \(\displaystyle B^{-1} - A^{-1} \) is also invertible.

For this one I did as follow:

\(\displaystyle ( B^{-1} - A^{-1} ) \cdot ( B^{-1} - A^{-1} )^{-1} = I_n \Leftrightarrow B \cdot ( B^{-1} - A^{-1} ) \cdot ( B^{-1} - A^{-1} )^{-1} = B \Leftrightarrow ( I_n - B \cdot A^{-1} ) \cdot ( B^{-1} - A^{-1} )^{-1} = B \Leftrightarrow \)

\(\displaystyle \Leftrightarrow ( I_n - B \cdot A^{-1} ) \cdot ( B^{-1} - A^{-1} )^{-1} \cdot ( B^{-1} - A^{-1} ) = B \cdot ( B^{-1} - A^{-1} ) \Leftrightarrow ( I_n - B \cdot A^{-1} ) \cdot I_n = I_n - B \cdot A^{-1} \Leftrightarrow \)

\(\displaystyle \Leftrightarrow I_n - B \cdot A^{-1} = I_n - B \cdot A^{-1} \).

Not sure if the solution is this, and I'd hope any one could correct me otherwise...
Also, my apologies if the LaTeX notations are not correct.

Thanks,
Diogo

 

[Romsek]

1) Is actually pretty straightforward when you apply method to it.

a) Assume \(\displaystyle A+B\) is invertible, noting that \(\displaystyle A\) is invertible show that \(\displaystyle (I_n +BA^{-1})A\) is invertible
and thus \(\displaystyle I_n + B A^{-1}\) is invertible

b) Assume \(\displaystyle I_n + B A^{-1}\) is invertible, and apply pretty much the same logic as in (a) to show \(\displaystyle A+B\) is invertible

 

[diogomgf]

1) Is actually pretty straightforward when you apply method to it.

a) Assume \(\displaystyle A+B\) is invertible, noting that \(\displaystyle A\) is invertible show that \(\displaystyle (I_n +BA^{-1})A\) is invertible
and thus \(\displaystyle I_n + B A^{-1}\) is invertible

b) Assume \(\displaystyle I_n + B A^{-1}\) is invertible, and apply pretty much the same logic as in (a) to show \(\displaystyle A+B\) is invertible

Thanks for the tip! That makes it alot easier

Cheers,
Diogo