Determinant of order 2 block matrix following specific instructions

Question

Given a block matrix of the form $$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}$$ with $A_{11}$ invertible, I want to prove that $$\det (A) = \det (A_{11}) \det \left(A_{22}-A_{21}A_{11}^{-1}A_{12}\right)$$ I know this has already been asked and answered several times. The problem is that the exercise says I should first compute the determinant of the following matrices: $$ A_1 = \begin{pmatrix} A_{11} & 0 \\ 0 & I \end{pmatrix} \qquad A_2 =\begin{pmatrix} I & 0 \\ 0 & A_{22} \end{pmatrix} \qquad A_3 = \begin{pmatrix} A_{11} & 0 \\ 0 & A_{22} \end{pmatrix} \qquad A_4 =\begin{pmatrix} I & A_{11}^{-1}A_{12} \\ 0 & I \end{pmatrix} \qquad A_5 =\begin{pmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{pmatrix} \qquad $$ This is obviously pretty easy, but from here on I don't see how to finish the proof. I know that I have to give a specific factorization of $A$ which allows me to calculate its determinant, but I don't know how to do it by using this five matrices.