Find the maximum number of complex numbers α for which the matrix $αA+B$ is not invertible

Question

Let $A$ and$ B$ be $(2017)^2 × (2017)^2$ complex matrices with $A $ being invertible. Find the maximum number of complex numbers α for which the matrix $αA+B$ is not invertible

If I take $A= I_{(2017)^2 × (2017)^2}$and $B= -I_{(2017)^2 × (2017)^2}$

now i get $ \alpha = 1$

Am I correct ??? is im on Right track ???

Answer 1

Hint:

$$\alpha A + B = A(\alpha I + A^{-1}B)=A(A^{-1}B-(-\alpha)I)$$

$\alpha A + B$ is not invertible if and only if $A^{-1}B-(-\alpha) I$ is not invertible, that is when the determinant is equal to $0$.

Think in terms of eigenvalues.

Remark:

For your attempt, you are right for your particular example of $A$ and $B$, but we have to work with general invertible $A$ and general $B$.

Answer 2

Following the same idea as @Siong we have $ \det(\alpha A+B)=\det(A)\det(\alpha I+BA^{-1})=0$ if and only if $\det(\alpha I+BA^{-1})=0$. Now by observing the degree of the polynomial $\det(x I+BA^{-1})$ we are through.