Classify each of the following matrices according as it is (a) real, (b) symmetric, (c) skew-symmetric, (d) Hermitian, or (e) skew-hermitian, and iden

Step 1
Real matrix $\Rightarrow A$ real matrix is a matrix whose elements consist entirely of real numbers.
Symmetric matrix $\Rightarrow A$ symmetric matrix is a square matrix that is equal to its transpose.
$A^T=A$
Skew symmetric matrix $\Rightarrow A$ matrix can be skew symmetric only if it is square. If the transpose of a matrix is equal to the negative of itself, the matrix is said to be skew symmetric.
$A^T=-A$
Step 2
Hermitian matrices $\Rightarrow$ Hermitian matrices can be understood as the complex extension of real symmetric matrices.
$A^{\theta }=A$
skew-Hermitian matrices $\Rightarrow$ skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers.
$A^{\theta }=-A$
Step 3
(1) Consider the provided question,
Let $A=\left[\begin{array}{ccc}1& 0& -i\\ 0& -2& 4-i\\ i& 4+i& 3\end{array}\right]$
Now, check the given matrix for the above condition.
check for Hermitian matrix,
$\overline{A}=\left[\begin{array}{ccc}1& 0& i\\ 0& -2& 4+i\\ -i& 4-i& 3\end{array}\right]$ Now $A^{\theta }={\overline{A}}^T=\left[\begin{array}{ccc}1& 0& -i\\ 0& -2& 4-i\\ i& 4+i& 3\end{array}\right]$
Therefore , $A^{\theta }=A$
So, it is satisfy the Hermitian matrix,
Step 4
The principal diagonal element of the matrix,
$\left[\begin{array}{ccc}1& 0& -i\\ 0& -2& 4-i\\ i& 4+i& 3\end{array}\right]$ is $\left[\begin{array}{ccc}1& -2& 3\end{array}\right]$
The Secondary diagonal element of the matrix,
$\left[\begin{array}{ccc}1& 0& -i\\ 0& -2& 4-i\\ i& 4+i& 3\end{array}\right]$ is $\left[\begin{array}{ccc}-i& -2& i\end{array}\right]$
Step 5
(2) Consider the provided question,
Let $A=\left[\begin{array}{ccc}7& 0& 4\\ 0& -2& 10\\ 4& 10& 5\end{array}\right]$
Now, check the given matrix for the above condition.
check for symmetric matrix,
$A^T=\left[\begin{array}{ccc}7& 0& 4\\ 0& -2& 10\\ 4& 10& 5\end{array}\right]$ therefore , $A^T=A$
So, it is satisfy the symmetric matrix,
Step 6
The principal diagonal element of the matrix,
$\left[\begin{array}{ccc}7& 0& 4\\ 0& -2& 10\\ 4& 10& 5\end{array}\right]$ is $\left[\begin{array}{ccc}7& -2& 5\end{array}\right]$
The Secondary diagonal element of the matrix,
$\left[\begin{array}{ccc}7& 0& 4\\ 0& -2& 10\\ 4& 10& 5\end{array}\right]$ is $\left[\begin{array}{ccc}4& -2& 4\end{array}\right]$

Answer is given below (on video)