Introduction to Matrices 🧮

A matrix is a grid of numbers or mathematical expressions that we use to store vast quantities of data in such a way that allows us to easily identify and manipulate each piece of information in the matrix. Engineering, science and computing all use matrices in a huge number of applications i.e. data analysis, search engines or game theory. The size of the matrices is observed by the number of rows and columns and can be described as an “ $m \; n \;$ ” matrix and written as ${m}\cdot{n}$ .

Example 1:

The matrix shown is a ${2}\cdot{4}$ matrix.

$A = \begin{bmatrix}3 & 0 & 9 & \pi \\ x & -6 & 2 & 4j \end{bmatrix}$

The individual values in the matrix are referred to as expressions and are usually written in lowercase letters. The $i$ th row of the $j$ th column can be written as $a_{ij}$ . The row is always stated first and then the column second.

$b_{11}$ is the first row of the first column, so from example 1 above $b_{11} = 3$

$b_{23}$ is the second row of the second column, so from example 1 above $b_{23} = 2$

$b_{14}$ is the first row of the fourth column, so from example 1 above $b_{14} = \pi$

Special Matrices

Square Matrices Examples:

Matrices that have the same number of rows as they do columns are referred to as a square matrix.

${2}\cdot{2}$

$\begin{bmatrix}4 & 7 \\ x & -6 \end{bmatrix}$

${3}\cdot{3}$

$\begin{bmatrix} 4 & 7 & -3 \\ 4x & -5 & 4 \\ \pi & 2x & -7 \end{bmatrix}$

${4}\cdot{4}$

$\begin{bmatrix} 1 & 2 & -3 & 4 \\ a & b^2 & c & d \\ \pi & 2x & -7 & 1 \\ 18 & 3 & -7j & 1k \end{bmatrix}$

Diagonal Matrices Examples:

Matrices that have zeros covering the entire matrix, apart from a line of expressions running from top-left to bottom-right of the matrix are called diagonal matrices as seen below.

$\begin{bmatrix}4 & 0 \\ 0 & -6 \end{bmatrix}$

$\begin{bmatrix} 4 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -7 \end{bmatrix}$

$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & b^2 & 0& 0 \\ 0 & 0 & -7 & 0 \\ 0 & 0 & 0 & 1k \end{bmatrix}$

Note: the middle value in the $3\cdot3$ matrix is a zero; yet this still qualifies as a diagonal matrix.

Identity Matrices Examples:

If there are all ones in the top left to bottom right expressions, and all the other values in the matrix are zero, then we have an identity matrix. Identity matrices will usually be denoted with an $I$ .