# Math Help - Square matrices within a matrix

1. ## Square matrices within a matrix

Prove that if M is a n x n matrix that can be written in $M = \left({\begin{array}{cc} A & B \\ 0 & C \end{array}} \right)$

where A and C are square matrices, then det(M) = det(A) det(C)

2. Write the matrix $A$ in Jordan normal form with $\lambda_1, ..., \lambda_k$ on the diagonal and 0's below the diagonal. Then $\text{det}(A) = \lambda_1 \lambda_2 ...\lambda_k.$ Likewise matrix $C$ can be written in this form and we know $\text{det}(C) = \lambda_{k+1}\lambda_{k+2}...\lambda_n$.

But then the entire matrix $M$ is in Jordan normal form with $\lambda_1, ..., \lambda_n$ on the diagonal and 0's below the diagonal, thus $\text{det}(M) = \lambda_{1}\lambda_{2}...\lambda_n = \text{det}(A)\text{det}(C)$.