Matrix Multiplication

matrix_multiplication

Let $A$ be a $m\times n$ matrix, and $B$ be a $n\times p$ matrix. The product is $AB$ an $m\times p$ matrix, equal to: $AB=A\left({\mathbf{b}}_1\dots {\mathbf{b}}_{\mathbf{p}}\right)=\left(A{\mathbf{b}}_1\dots A{\mathbf{b}}_{\mathbf{p}}\right)$

\begin{aligned} AB &= \left[ \begin{array} \\ 1 & 0\\1&1 \end{array} \right] \left[ \begin{array} \\ 2 & 0 &0\\3&4&0 \end{array} \right] \\ &=\left( \left[ \begin{array} \\ 1 & 0\\1&1 \end{array} \right] \left[ \begin{array} \\ 2 \\ 3\end{array} \right] \ \ \ \ \left[ \begin{array} \\ 1 & 0\\1&1 \end{array} \right] \left[ \begin{array} \\ 0 \\4 \end{array} \right] \ \ \ \ \left[ \begin{array} \\ 1 & 0\\1&1 \end{array} \right] \left[ \begin{array} \\0\\0 \end{array} \right] \right)\\ &=\left[ \begin{array} \\ 2&0&0\\5&4&0 \end{array} \right] \end{aligned}

(Note: This is very similar to Matrix Vector Product)

Important!