Define variables a and b in matrix with determinant

Question

I should calclulate the determinant of a Matrix (4x4) and define the pairs if $a$ and $b \in R$.

I calculated this determinant: $ab\left(a-2b\right)\left(2a+b\right),$ but how can I use the determinant to define $a$ and $b$ such that A is invertible?

Answer 1

Recall inverse of a matrix = $\frac{1}{det(A)}adj(A)$

So, for invertible matrix $ab\left(a-2b\right)\left(2a+b\right)\neq0$

Therefore $a\neq 0 , b\neq 0, a\neq 2b \text{ and } b\neq −2a. $

Answer 2

$A$ is invertible $ \iff ab\left(a-2b\right)\left(2a+b\right) \ne 0.$

Can you proceed ?