3
$\begingroup$

If $A$ and $B$ are any $3 \times 3$ matrices and A is any invertible matrix, then there exist an integer $n$ such that $A + nB$ is invertible.

It is easy to check if we take $n = 0$, then the result always holds, But I want to know, when $n$ is non-zero then the result is true or not.

  • 0
    Wh$y$ did you accept an incomplete solution?2012-09-30

1 Answers 1

3

Yes, the determinant of $A+nB$ can be written as a polynomial $f$ in $n$ of degree 3. There are at most 3 real roots of $f$ and any integer $m$ which is not a root gives $A+mB$ which has non-zero determinant, and so invertible.

  • 0
    @JoelCohen **This** is a proof.2012-09-30