##### Lecture on Symmetric, Skew Symmetric and Orthogonal Matrix
Content covered:

SYMMETRIC, SKEW SYMMETRIC AND ORTHOGONAL MATRIX
In a Symmetric matrix $A=A^t$ iff, A is a square matrix

$A=\begin{bmatrix}a&b\\b&c\end{bmatrix}$ ,  $A^t=\begin{bmatrix}a&b\\b&c \end{bmatrix}$

Properties of symmetric matrix:

1. $AA^t$ is a symmetric matrix
2. $(A+A^T)/2$ is a symmetric matrix

Skew Symmetric Matrix

A matrix is skew symmetric matrix iff, $A^T=-A$

$A=\begin{bmatrix}a&e&d\\-e&b&f\\-d&-f&c\end{bmatrix}$

$A^T=\begin{bmatrix}a&-e&-d\\e&b&-f\\d& f&c\end{bmatrix}$

= -A

$tr(A)=tr(A^t )$

Properties of skew symmetric matrix:

1. $( A-A^T)/2$
is a skew symmetric matrix.
2. For, skew symmetric matrix, diagonal elements will be equal to 0.

In Orthogonal Matrix , $A^T=A^-1$

$AA^-1=I$

$|AA^-1|=|I|$

$(|A|)^2=I=1$