# Symmetric and Skew Symmetric Matrices

• Symmetric Matrix : A square matrix A is said to be symmetric if AT = 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 AT = – 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 BT AB is symmetric.
• If A is skew symmetric matrix and C is non zero column matrix then CT 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 (Aθ) : The transpose of a conjugate of a matrix A is called transposed conjugate of A and it is denoted by Aθ.
• Hermitian matrix : A square matrix A is said to be hermitian if Aθ = A
• Skew hermitian matrix : A square matrix A is said to be skew hermitian if Aθ = – 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 2n.
• The number of all possible matrices of order n x n where entries are 1, 2, 3, ........ n with out repetation is n!