Proving properties of triangular matrices

Question

This is the question that I'm having trouble with:

I understand what the alternating and multilinear properties are, and I know that the determinant of the matrix is an alternating, multilinear function of the coloumns. But I don't understand the exact significance in this question.

I'm assuming that I cannot simply say that the LHS = det(A)det(B) = RHS ? Surely it must be more complicated than this?

Answer 1

You are expected to use the property that if $f(A)$ is a multilinear and alternating functio of the columns of $A$, then $f(A)=\det(A)f(I)$. Now, take $f(A)=\det\pmatrix{A&C\\0 &B}$.