## Matrices

# Symmetric and Skew Symmetric Matrices

**Symmetric Matrix :**A square matrix A is said to be symmetric if A^{T}= A

Ex: \tt A=\begin{bmatrix}1 & 2 & 3 \\2 & 4 & 6 \\3 & 6 & 5 \end{bmatrix}**Skew Symmetric Matrix :**A square matrix A is said to be Skew Symmetric if A^{T}= – A

Ex : \tt A=\begin{bmatrix}0 & 1 & 2 \\-1 & 0 & -3 \\-2 & 3 & 0 \end{bmatrix}- All positive integral powers of symmetric matrix is symmetric.
- All positive odd integral powers of skew symmetric matrix is skew symmetric.
- All positive even integral powers of skew symmetric matrix is symmetric.
- If A is both symmetric and skew symmetric then it is null matrix.
- If A and B symmetric matrices of same order then AB + BA is symmetric.
- If A and B symmetric matrices of same order then AB – BA is skew symmetric.
- If A and B are square matrices of same order and A is symmetric then B
^{T}AB is symmetric. - If A is skew symmetric matrix and C is non zero column matrix then C
^{T}AC is null matrix. **Conjugate of a matrix :**The matrix is obtained from a matrix A on replacing its elements by the corresponding conjugate complex numbers is called the conjugate of A.**Transposed conjugate of A (A**The transpose of a conjugate of a matrix A is called transposed conjugate of A and it is denoted by A^{θ}) :^{θ}.**Hermitian matrix :**A square matrix A is said to be hermitian if A^{θ}= A**Skew hermitian matrix :**A square matrix A is said to be skew hermitian if A^{θ}= – A.

### View the Topic in this video From 19:45 To 36:26

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- Every square matrix can be express as sum of symmetric and skew symmetric matrices ie.
- The maximum number of distinct elements in symmetric matrix of order n is \tt \frac{n\left(n+1\right)}{2}
- If \tt A=\begin{bmatrix}\cos \theta & \sin \theta \\-\sin \theta & \cos \theta \end{bmatrix}then \ A^{n}=\begin{bmatrix}\cos n\theta & \sin n \theta \\-\sin n \theta & \cos n \theta \end{bmatrix}\forall \ n \in N.
- If \tt A=\begin{bmatrix}a & 0 & 0 \\0 & b & 0\\0 & 0 & c \end{bmatrix}then \ A^{n}=\begin{bmatrix}a^{n} & 0 & 0 \\0 & b^{n} & 0\\0 & 0 & c^{n} \end{bmatrix}\forall \ n \in N
- The number of all possible matrices of order n x n whole entrices are either 0 (or) 1 is 2
^{n}. - The number of all possible matrices of order n x n where entries are 1, 2, 3, ........ n with out repetation is n!