- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
3.11. $\quad$ Exercises
6. $\quad$ State and prove a generalization of Exercise 5 for $n \times n$ matrices of the form $$A = \begin{bmatrix} B & O \\ C & D \end{bmatrix}$$ where $B,$ $C,$ $D$ denote square matrices and $O$ denotes a matrix of zeros.
Solution. $\quad$ As shown in Exercise 5, we can use the row operations of the Gauss-Jordan elimination process to transform $A$ into the equivalent block-diagonal matrix $A = \begin{bmatrix} B & O \\ O & D \end{bmatrix},$ whose determinant is given by \begin{align*} \det A &= (\det B)(\det D). \quad \blacksquare \end{align*}