$2\times 2$ Invertible Matrices in the Field Mod $2$?

Question

Which $2\times 2$ matrices in the field mod $2$ are in $GL(2,\Bbb F)$ (i.e. have an invertible $2\times 2$ matrix in the same field which multiply to give you the identity matrix)? I could go case by case but was hoping there was a nicer approach.

Answer 1

A square matrix of any dimension with elements in any commutative, unital ring is invertible if its determinant is invertible. This is because for any square matrix $M$, you can define the adjugate matrix $\operatorname{adj}(M)$ with the property that $$ \operatorname{adj}(M)\,M = M\,\operatorname{adj}(M) = \det(M)\cdot I $$ which means that $M^{-1} = \det(M)^{-1}\operatorname{adj}(M)$, if the determinant is invertible.