I am trying to build two non-zero square matrices $A$ and $B$ whose product will be zero and who will have any fixed determinant value (e.g. det$(A) = 5$).
I can easily think of two non-zero square matrices that satisfy $AB = 0$, but to get them to have a specific determinant is tripping me up.
Would anyone know of a first step? I imagine it would be easy to start with two triangular matrices.
Hint:
If $A \ne 0$ and $B\ne 0$ are such that $AB=0$, than $A$ and $B$ are not invertible and this means that $\det A =0$ and $\det B=0$.
You can prove this by contraposition. Suppose $A$ is invertible, than
$$ AB=0 \Rightarrow A^{-1}(AB)=A^{-1}0 \Rightarrow (A^{-1}A)B=0\Rightarrow B=0 $$
and analogously for $B$.