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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 , \(A^T=A^-1\)

\(AA^-1=I\)

\(|AA^-1|=|I|\)

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