This is a question from TIFR GS $2011$.But I think this is a stupid question or the question is not precise.This question asks if given $A,B$ matrices of order $3\times 3$ ,and $A$ invertible,there exists any $n\in \mathbb Z$ such that $A+nB$ is invertible.My simple answer would be yes,because $n=0\in \mathbb Z$ and $A+nB=A+0B=A$ which is of course invertible as $A$ is invertible.
Can someone help me to figure out if this question is stupid or the question should be made precise.If it the latter case,then one may write the precise version.I am also looking for a proof of this problem with linear transformations.
Also can someone suggest similar or more general results related to this problem?
There are at least two issues with this question. You raised the first one: $n$ should be a non-zero integer, otherwise the answer is trivial. The other issue is that the ring of the coefficients is not given. In particular, if the characteristic of the ring is $n$, then any multiple of $n$ is a trivial solution, and thus it would also be better to discard these trivial solutions. But in this case, there might be no solution. For instance, if the ring is the field ${\Bbb F}_2$ and $A$ and $B$ are the identity matrix, then $A + nB$ is invertible if and only if $n$ is even.