There exists at most $n$ real numbers such that $tA+B$ is not invertible

Question

A matrix like $A \in M_n(F)$ is invertible iff there exists another matrix like $D \in M_n(F)$ such that $AD=I_n$.

The question :
Assume that $A,B \in M_n(\mathbb R)$ and $A$ is invertible.
Prove that there exists at most $n$ real numbers like $t$ such that $tA+B$ is not invertible.

Note : My problem is that i can't just try every real number. Also, there exists another question similar to mine but doesn't have my answer.

Answer 1

You have $$tA+B=A (tI+A^{-1}B). $$So $tA+B $ will be singular precisely when $tI+A^{-1} $ is singular. Then $t $ is necessarily an eigenvalue of $A^{-1}B $.