Teach me matrix on mathematics

A matrix is typically denoted by a capital letter. The order or dimension of a matrix is given by the number of rows by the number of columns ($m \times n$).

For example, a matrix $A$ with $m$ rows and $n$ columns looks like this:

$$A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$$

Here, $a_{ij}$ represents the element in the $i$-th row and $j$-th column.

Consider the following matrix $B$:

$$B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$

This matrix has 2 rows and 3 columns. Therefore, its order is $2 \times 3$.
For example, the element $a_{12}$ (first row, second column) is 2, and $a_{21}$ (second row, first column) is 4.