Product of Partitioned Matrices

Theorem

Let

• $A=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]$
• $B=\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right]$

Then

$$\begin{array}{rl}AB& =\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right]\\ & =\left[\begin{array}{cc}{A}_{11}{B}_{11}+A_{12}{B}_{21}& A_{11}{B}_{12}+A_{12}{B}_{22}\\ A_{21}{B}_{11}+A_{22}{B}_{21}& A_{21}{B}_{12}+A_{12}{B}_{22}\end{array}\right]\end{array}$$

Tip

The pattern is the same as matrix product with real numbers.