Question: Give numerical examples of: a symmetric matrix; a skew-symmetric matrix; a real matrix; a pure imaginary matrix.

Symmetric matrix:

The elements present on both sides of diagonal elements are same.

Skew symmetric matrix:

The diagonal elements of matrix are zero and rest elements follows like.

${\mathbf{a}}_{\mathbf{32}}\mathbf{=}\mathbf{-}{\mathbf{a}}_{\mathbf{23}}$

Real matrix:

The diagonal elements are unit and rest elements follow symmetry for e.g.

${\mathbf{a}}_{\mathbf{12}}\mathbf{=}{\mathbf{a}}_{\mathbf{21}}$

Pure imaginary matrix:

All elements of a pure imaginary matrix are imaginary numbers.

The examples of symmetric, skew symmetric, real and pure imaginary matrices are given below:

Symmetric matrix:

The elements present on both sides of diagonal elements are same for e.g.

${\mathrm{a}}_{21}={\mathrm{a}}_{12},{\mathrm{a}}_{32}={\mathrm{a}}_{23}$

Example:

$$\begin{gathered} \left(168 \\ 627 \\ 873\right) \end{gathered}$$

Skew symmetric matrix:

The diagonal elements of matrix are zero and rest elements follows like.

Example:

$$\begin{gathered} \left(095 \\ -903 \\ -5-30\right) \end{gathered}$$

Real matrix:

The diagonal elements are unit and rest elements follow symmetry for e.g.

Example:

$$\begin{gathered} \left(198 \\ 917 \\ 871\right) \end{gathered}$$

Pure imaginary matrix:

All elements of a pure imaginary matrix are imaginary numbers.

Example:

$$\begin{gathered} \left(\mathrm{i}5+\mathrm{i} \\ -5+\mathrm{i}8\mathrm{i}\right) \end{gathered}$$