Matrix Multiplication

Blocks §

$$\left[\begin{array}{cc}{A}_1& A_2\\ A_3& A_4\end{array}\right]\left[\begin{array}{cc}{B}_1& B_2\\ B_3& B_4\end{array}\right]=\left[\begin{array}{cc}{A}_1{B}_1+A_2{B}_3& A_1{B}_2+A_2{B}_4\\ A_3{B}_1+A_4{B}_3& A_3{B}_2+A_4{B}_4\end{array}\right]$$

Linear Combination §

A matrix times by a vector is just a linear combination of the column vectors of the matrix.

$$\begin{array}{r}\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=1\left[\begin{array}{c}2\\ -1\end{array}\right]+2\left[\begin{array}{c}-1\\ 2\end{array}\right]=\left[\begin{array}{c}0\\ 3\end{array}\right]\end{array}$$