Matrix

==2024-11-20

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Theory

It is a very disciplined notion like sets.

Defination

A rectangular arrangement of elements (numbers) into rows and columns.

Set for Matrix

The set for a matrix of order $m\times n$ is:

$$M_{m\times n}(\mathbb{F}(\mathbb{R}/\mathbb{C},+,.))$$

Symbolically a matrix $A=[a_{ij}]$ of order $m\times n$ is given by,

$$A=[a_{ij}{]}_{m\times n}=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & a_{2n}\\ a_{31}& a_{32}& a_{33}& \dots & a_{3n}\\ ⋮& ⋮& ⋮& a_{ij}& ⋮\\ a_{m1}& a_{m_2}& a_{m_3}& \dots & a_{mn}\end{array}\right]$$

if $a_{ij}\in \mathbb{R}\to A$ is real matrix. if $a_{ij}\in \mathbb{C}\to A$ is complex matrix. But since, $\mathbb{R}\subset \mathbb{C}$, every real matrix is a complex matrix.

$a_{ij}$ denotes the element at $i^{th}$ row & $j^{th}$ column.

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Examples

$$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$$

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Implementation

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PTR