6.2 Basics

Matrix algebra often proves useful. We assume you recall how to add, multiply and do basic matrix operations.

Example 6.5 (Matrix addition) Consider the matrices: $A = \left[ \begin{array}{rrr} 1 & 11 & 21\\ 2 & 12 & 22\\ \end{array} \right] \quad\text{and}\quad {B} = \left[ \begin{array}{rrr} -1 & 4 & -5\\ 0 & 3 & 2\\ \end{array} \right].$ Then $A + B$ is $A + B = \left[ \begin{array}{rrr} 1 + (-1)& 11 + 4 & 21 + (-5)\\ 2 + 0 & 12 + 3 & 22 + 2\\ \end{array} \right] = \left[ \begin{array}{ccc} 0 & 15 & 16\\ 2 & 15 & 24\\ \end{array} \right].$

Example 6.6 (Matrix multiplication) Using the same matrices $A$ and $B$ as above, we cannot find $A \times B$ as the sizes do not conform. However, $\begin{aligned} A^T \times B &= \left[ \begin{array}{rr} 1 & 2\\ 11 & 12\\ 21 & 22 \end{array} \right] \times \left[ \begin{array}{rrr} -1 & 4 & -5\\ 0 & 3 & 2 \end{array} \right]\\ &= \left[ \begin{array}{rrr} (\phantom{0}1\times -1) + (\phantom{0}2\times 0) & (\phantom{0}1\times 4) + (\phantom{0}2\times 3) & (\phantom{0}1\times -5) + (\phantom{0}2\times 2)\\ (11\times -1) + (12\times 0) & (11\times 4) + (12\times 3) & (11\times -5) + (12\times 2)\\ (21\times -1) + (22\times 0) & (21\times 4) + (22\times 3) & (21\times -5) + (22\times 2)\\ \end{array} \right]\\ &= \left[ \begin{array}{rrr} -1 & 10 & -1 \\ -11 & 80 & -31\\ -21 & 150 & -61 \\ \end{array} \right]. \end{aligned}$

It is worth remembering that matrix multiplication can only occur between two conformable matrices. That is, consider matrix $A$ of size $n\times m$ and matrix $B$ of size $m\times p$ . Then we can multiply $A\times B$ , producing a $(n\times m) \times (m\times p) = n\times p$ matrix, but we cannot multiply $B$ by $A$ .