Citation for block matrix determinant formulae

Question

The wikipedia page on determinants has the following equality:

When $\mathbf{A}$ is invertible one has

$$\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(A) \det\left(D - C A^{-1} B\right)$$

However, the citation for this factoid is an online manual. Does anyone know of a textbook that contains this formula?

Answer 1

For matrix $M = \begin{bmatrix}A & B\\ C & D\end{bmatrix}$, the expression

$$M/A \stackrel{def}{=} D - CA^{-1}B$$ is called the Schur complement of block $A$ of matrix $M$. For invertible $A$, the relation

$$\det M = \det(A) \det(M/A) = \det(A)\det(D - CA^{-1}B)\tag{*1}$$

first appear in a proof of following lemma in a paper by Schur ${}^\color{blue}{[1]}$.

Schur determinant lemma
Let $M = \begin{bmatrix}P & Q\\ R & S\end{bmatrix}$ be a $2n \times 2n$ matrix partitioned into $n\times n$ blocks $P, Q, R, S$. If $P$ commutes with $R$, then $\det M = \det(PS - RQ)$

Some author refer to identity $(*1)$ as Schur's Formula. For more details on Schur's complement and related properties/applications, please consult Zhang's book ${}^{\color{blue}{[2]}}$.

References

• $\color{blue}{[1]}$ - Schur, I., Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind [I], Journal für die reine und angewandte Mathematik, 147 (1917), 205-232.
  A digitized copy of the paper can be found here. The formula appear at around page 217.

• $\color{blue}{[2]}$ - Zhang, Fuzhen (2005). The Schur Complement and Its Applications Springer. (see Theorem 1.1 there).

Answer 2

Note that $$\begin{pmatrix}I& 0\\ -CA^{-1}& I\end{pmatrix}\begin{pmatrix}A& B\\ C& D\end{pmatrix}=\begin{pmatrix}A& B\\ 0& D-CA^{-1}B\end{pmatrix}$$.

Then taking the determinates of both sides.