MHB Can Nonsingular Matrices Be Generated by Multiplying and Adding?

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Here is this week's POTW:

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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.

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This week's problem was solved correctly by Opalg. You can read his solution below.

Spoiler

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.