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 \(A\).