Introduction to Matrix

What is Matrix?

In mathematics, a matrix is a rectangular array or table of numbers, arranged in rows and columns, and enclosed by brackets ( ) or square brackets [ ], which is used to represent a mathematical object or a property of such an object. Numbers arranged in a matrix are called matrix's elements or entries.
Matrices are usually symbolized using upper-case letters, while the corresponding lower-case letters, with two subscript indices. Matrices are widely used in various fields, including mathematics, computer science, physics, and engineering, to represent and manipulate data.

Upper-case letters ($A$ - $Z$) is to represent the matrix itself.
Lower-case letters with subscripts ($a_{m,n}$ or $a_{mn}$) is to represent the entry indices.

$$\begin{matrix} \\ ^{m_1} \\ ^{m_2} \\ \vdots \\ ^{m_\infty} \end{matrix} \hspace{-0.2em} \begin{matrix} \\ \left [ \begin{matrix} \\ \\ \\ \\ \\ \end{matrix} \right. \end{matrix} \begin{matrix} _{n_1} & _{n_2} & \cdots & _{n_\infty} \\ \hline a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\\ a_{2,1} & a_{2,2} & \cdots & \vdots \\\ \vdots & \vdots & \ddots & \vdots \\\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{matrix} \begin{matrix} \\ \left. \begin{matrix} \\ \\ \\ \\ \\ \end{matrix} \right ] \end{matrix}$$

Matrix algebra is frequently implemented in geometry to express and represent geometric transformations (such as rotations as well as coordinate changes. Plenty of computational issues in numerical analysis are tackled by simplifying them to matrix computations, which frequently involve calculating with matrices of enormous scale. Matrices are employed in almost every branch of mathematics and science, either directly or indirectly through geometry and numerical analysis.

Matrix Row

Row matrix is arrangement of numbers (elements) that are arranged in a horizontally manner. A row matrix has only one row with multiple columns.
The row matrix order is $1 \times n$, and has $n$ columns.

$$\begin{bmatrix} a_{1,1} \\\ a_{2,1} \\\ a_{3,1} \\\ \vdots \\\ a_{m,1} \end{bmatrix} \hspace{3pt} \begin{matrix} \to \text{row } 1 \\\ \to \text{row } 2 \\\ \to \text{row } 3 \\\ \\\ \to \text{row } m \end{matrix}$$

Properties of Matrix Row

A row matrix consists of numerous columns.
A row matrix also acts as a rectangular matrix.
A column matrix is the transpose of a row matrix.
Only row matrices of similar order or size are allowed to operate addition or subtraction.
A row matrix is only capable of being multiplied by a column matrix.
The numerous columns are according to how many elements there are in the matrix.

Matrix Column

Column matrix is an arrangement of numbers (elements) that are arranged in a vertically manner. A column matrix has only one column with multiple rows.
The order of a column is $m \times 1$, and has $m$ rows.

$$\begin{matrix} \\ \left[ \right. \end{matrix} \begin{matrix} \text{column } 1 & \text{column } 2 & \text{column } 3 & \cdots & \text{column } n \\ \hline a_{1,1} & a_{1,2} & a_{1,3} & \cdots & a_{1,n} \end{matrix} \begin{matrix} \\ \left. \right] \end{matrix}$$

Properties of Matrix Column

A column matrix consists of numerous rows.
A column matrix also acts as a rectangular matrix.
A row matrix is the transpose of a column matrix.
Only column matrices of similar order or size are allowed to operate addition or subtraction.
A column matrix is only capable of being multiplied by a row matrix.
The numerous rows are according to how many elements there are in the matrix.

Matrix Order

The order of a matrix refers to the arrangement of elements in terms of the number of rows and columns present in the matrix, also known as the dimensions of the matrix.

If a matrix has $m$ rows and $n$ columns, its matrix order is represented as $m \times n$ (also read as "$m$ by $n$"), and it can be denoted as $A_{m,n}$.

For example, let's consider matrix below:

$$M = \begin{bmatrix} m_{1,1} & m_{1,2} & m_{1,3} & m_{1,4} & m_{1,5} \\\ m_{2,1} & m_{2,2} & m_{2,3} & m_{2,4} & m_{2,5} \end{bmatrix}$$

The matrix above can be written as $M_{2,5}$. This is because the matrix $M$ has $2$ rows and $5$ columns, as indicated by the subscript $2$ and $5$, respectively.