Symmetric and Skew Symmetric Matrices
Symmetric Matrix: A square matrix A is said to be a symmetric matrix if AT = A.
Examples:
1. \(A=\left[ \begin{matrix}2 & 3 & 1\\3 & 4 & 5\\1 & 5 & 7\\\end{matrix} \right]\).
\({{A}^{T}}=\left[ \begin{matrix}2 & 3 & 1\\3& 4 & 5\\1 & 5 & 7\\\end{matrix} \right]\).
Hence A = AT then A is symmetric matrices.
\(A=\left[ \begin{matrix}1 & -1\\-1 & 2\\\end{matrix} \right]\).
\({{A}^{T}}=\left[ \begin{matrix}1 & -1\\-1 & 2\\\end{matrix} \right]\).
Hence A = AT then A is symmetric matrices.
3. If A is a square matrix then show that A + AT, AAT are symmetric matrices.
Solution:
(A + AT)T
= AT + (AT)T
= AT + A
= A + AT
→ A + AT is a symmetric matrix.
(AAT)T = (AT)T
AT = AAT → AAT is symmetric matrix.
Note: If A is a symmetric matrix and k is a scalar then kA is a symmetric matrix.
Skew Symmetric Matrix: A square matrix A is said to be a skew symmetric matrix if AT = -A.
Examples:
1. \(A=\left[ \begin{matrix}0 & 1 & -2\\-1 & 0 & 3\\2 & -3 & 0\\\end{matrix} \right]\).
\({{A}^{T}}=\left[ \begin{matrix}0 & -1 & 2\\1 & 0 & -3\\-2 & -3 & 0\\\end{matrix} \right]\).
\(-{{A}^{T}}=-\left[ \begin{matrix}0 & -1 & 2\\1 & 0 & -3\\-2 & -3 & 0\\\end{matrix} \right]\).
\(-{{A}^{T}}=\left[ \begin{matrix}0 & 1 & -2\\-1 & 0 & 3\\2 & 3 & 0\\\end{matrix} \right]\).
Hence A=-A T , then A is skew symmetric matrices.
\(A=\left[ \begin{matrix}0 & 5\\-5 & 0\\\end{matrix} \right]\).
\({{A}^{T}}=\left[ \begin{matrix}0 & -5\\5 & 0\\\end{matrix} \right]\).
\(-{{A}^{T}}=-\left[ \begin{matrix}0 & -5\\5 & 0\\\end{matrix} \right]\).
\(-{{A}^{T}}=\left[ \begin{matrix}0 & 5\\-5 & 0\\\end{matrix} \right]\).
Hence A = A T, then A is are skew symmetric matrices.
3. If A is a square matrix then show that A – AT is an skew symmetric matrix.
Solution:
(A – AT)T = AT
= – (AT)T
= AT
= – A
= -(A – AT) → A – AT is an skew symmetric matrix.
Note: If A is an skew symmetric matrix and k is a scalar then kA is an skew symmetric matrix.
Note: A skew symmetric matrix of order 3 is of the form
\(\left[ \begin{matrix}0 & a & b\\-a & 0 & c\\-b & -c & 0\\\end{matrix} \right]\).