Welcome to our community

Be a part of something great, join today!

Problem of the Week #219 - Aug 09, 2016

Status
Not open for further replies.
  • Thread starter
  • Moderator
  • #1

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
Here is this week's POTW:

-----
Let $A$ and $B$ be nonsingular $n\times n$-matrices over a field $\Bbb k$. Show that for all but finitely many $x\in \Bbb k$, $xA + B$ is nonsingular.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!
 
  • Thread starter
  • Moderator
  • #2

Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,892
This week's problem was solved correctly by Opalg . You can read his solution below.


If $A$ is nonsingular then it has an inverse $A^{-1}$, and $xA+B = (xI + BA^{-1})A$. The matrix $BA^{-1}$ has at most $n$ distinct eigenvalues. If $-x$ is not one of those eigenvalues then $xI + BA^{-1}$ is invertible.

The product of two invertible matrices is invertible. Therefore if $-x$ is not an eigenvalue of $BA^{-1}$ then $xA+B$ is invertible. Hence there are only finitely many values of $x$ for which $xA+B$ is singular.
 
Status
Not open for further replies.