Question

What is a symmetric and skew-symmetric matrix?

Answer 1

Symmetric and skew symmetric matrix:

A matrix is symmetric if and only if it is equal to its transpose.

All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal.

If matrix $A$ is a symmetric matrix. Then,

$A=A^T$

A matrix is skew-symmetric if and only if it is the opposite of its transpose.

All main diagonal entries of a skew-symmetric matrix are zero.

If the matrix $A$is a skew-symmetric matrix. Then,

$A=-A^T$

Hence, If matrix $A$ is a symmetric matrix. Then, $A=A^T$ and If the matrix $A$is a skew-symmetric matrix. Then, $A=-A^T$.