Finding a Matrix from Determinants

Question

I've stumbled upon this problem on my homework, and I have no clue how to do it, and haven't found any help online:

If I'm understanding this correctly, then $det(M) = ad - cb + eh - gf$ ? What I don't get is how to find M from knowing this information.

Are there any suggestions on how to solve this?

Answer 1

Consider $\det(M)=ad-bc+eh-fg=(ad+eh)\times1-(bc+fg)\times1$ (assuming M is required to be 2x2, otherwise the trivial example by Jared in the comments would suffice)

Answer 2

It's unclear what the dimension of $M$ should be, but here is a start to a solution where $M$ is a $2\times 2$ matrix:

$$\begin{pmatrix}a&??\\1&d\end{pmatrix}$$

Try to finish it from here (i.e., find what goes in the entry labeled $??$).