Suppose we are given a matrix $X ∈ R^{a×a}$ of the form $X\:=\:\begin{pmatrix}A&B\\ 0&D\end{pmatrix}$ where $A ∈ R^{b×b}, B ∈ R^{b×(a−b)}$ and $D ∈ R^{(a−b)×(a−b)}$ for some $b ∈ {1,...,a}$.
Show that $\det(X) = \det(A) \det(D)$
This is solvable through Laplace but isn't there a trivial solution using Gaussian elimination? Since you can multiply the determinant of the diagonals and Gaussian can be applied separately to $A$ and $D$ since they are square matrices. However, my professor said this is iffy and no one could come up with an agreement on this proof method. What is missing or what would make this "trivial" proof sufficient and complete?