Apologies for this elementary question; but I was unable to find a reference otherwise.


Let $A, B, C$ be square matrices of the same dimension. Then, 


$$\begin{vmatrix} A & C \\\ 0 & B \end{vmatrix} = |A||B|.$$


The above statement is quite easy to prove using linearity etc properties of the determinant. However this statement is supposed to be a special case of a general theorem of Lagrange(the existence of which I read about, in Artin's Galois theory book).


So what is this general theorem of Langrange, and could someone please provide a reference? Most stuff is usually in Lang's Algebra, but not this one.



Statement of Lagrange's theorem on determinants(elementary question).
Let $A\_{S,T}$ denote the submatrix of $A$ with rows indexed by the elements of $S$ and columns
by the elements of $T$; let $A'\_{S,T}$ denote the submatrix of $A$ with rows indexed by the 
elements not in $S$ and columns by the elements not in $T$. Then we have an expansion
$$
 \det(A) = \sum\_T (-1)^{\omega(S,T)} \det(A\_{S,T})\det(A'\_{S,T})
$$
where $\omega(S,T)=|S|+|T|$ and $S$ is a fixed $k$-subset of the rows of $A$ and $T$ runs over
all $k$-subsets of the columns.


As you can see this makes short work of your identity. If you can prove the usual row expansion using exterior algebra, you can prove this generalization. I do not know a reference, but this expansion will be in any of the classical books on determinants
(possibly ascribed to Laplace). It can be used to provide a nice proof of the Pluecker relations, satisfied by the $n\times n$ minors of an $n\times 2n$ matrix.


