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.


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 $.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.