Which Matrix is must be Non-Singular?

Question

Let A and B be matrices such that B^2+AB+2I = 0,where I is an identity matrix.Which of the following matrices must be non-singular ? i) A ii) B iii) A+2I iv) B+2I

Answer 1

So we have $B^2+AB=-2I$. Thus $(B+A)B=-2I$. And so we know an inverse to $B$ is $\frac{-1}{2}(B+A)$, that is to say $B$ is invertible.