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Let $A$ be a $k\times k$ matrix and let $B$ be an $n\times n$ matrix. Let $C$ be $k\times n$. (All with entries in some field $K$). Suppose $D=\begin{pmatrix}C & A \\ B & 0\end{pmatrix}$. I want to find the determinant of $D$. I assume the way to go about this is using the Leibniz formula but I'm not sure where to start with that. What do I need to do?

MHW
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    Consider using row operations to transform $D$ to the block triangular matrix $\left( \begin{array}{cc} B & 0 \ C & A\end{array}\right)$. – DCarter Oct 25 '16 at 19:55
  • Following up on DCarter's suggestion, check http://math.stackexchange.com/a/75362/11069 – Sasha Oct 25 '16 at 20:06
  • @DCarter How would one go about this? – MHW Oct 25 '16 at 20:23
  • @sksks52 Exchanging two rows of a matrix multiplies determinant by (-1). Using $k$ row exchanges, we can transform your matrix into the block triangular matrix I have written; their determinants differ by the factor $(-1)^k$. And the determinant of a block triangular matrix is the product of the determinants of the diagonal blocks. – DCarter Oct 25 '16 at 20:29
  • @sksks52 I didn't say that correctly. Each "operation" I have in mind is a cyclic permutation of the rows that moves the first row to the bottom and pushes the other rows up. Each of these more complicated operations can be achieved using $k+n-1$ exchanges of adjacent rows. And each of these atomic exchanges changes determinant by the factor $(-1)$. So you'll need to do a little figuring to get the final sign correct. – DCarter Oct 25 '16 at 20:43

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