Symmetric Matrix & Skew Symmetric Matrix - Practice Questions & MCQ

Symmetric matrix:

A square matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}}$ is said to be symmetric if $\mathrm{A}^{\prime}=\mathrm{A}$,

$
\begin{aligned}
& \text { i.e., } \mathrm{a}_{\mathrm{ij}}=\mathrm{a}_{\mathrm{ji}} \forall \mathrm{i}, \mathrm{j} \\
& \mathrm{~A}=\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right] \text { then } \mathrm{A}^{\prime}=\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right]
\end{aligned}
$

Clearly, A = A', hence A is a symmetric matrix

Skew-symmetric matrix:  

A square matrix $A=\left[a_{i j}\right]_{m \times n}$ is said to be skew-symmetric if $A^{\prime}=-A$

$
\text { i.e. } \mathrm{A}^{\prime}=-\mathrm{A} \text {, i.e., } \mathrm{a}_{\mathrm{ij}}=-\mathrm{a}_{\mathrm{ji}} \forall \mathrm{i}, \mathrm{j}
$
Now if we put $\mathrm{i}=\mathrm{j}$, we have

$
\begin{aligned}
& \mathrm{a}_{\mathrm{ii}}=-\mathrm{a}_{\mathrm{ii}} \\
& \therefore 2 \mathrm{a}_{\mathrm{ii}}=0 \Rightarrow \mathrm{a}_{\mathrm{ii}}=0 \forall \mathrm{i}^{\prime} \mathrm{s}
\end{aligned}
$
That means all the diagonal elements of a skew-symmetric matrix are 0 .
e.g. $\mathrm{A}=\left[\begin{array}{ccc}0 & h & g \\ -h & 0 & f \\ -g & -f & 0\end{array}\right]$, then $\mathrm{A}^{\prime}=\left[\begin{array}{ccc}0 & -h & -g \\ h & 0 & -f \\ g & f & 0\end{array}\right]=-\mathrm{A}$

Properties of Symmetric and Skew-symmetric Matrices:

 i) If $A$ is a square matrix, then $A A^{\prime}$ and $A^{\prime} A$ are symmetric matrices
ii) If $A$ is a symmetric matrix, then $-A, k A, A^{\prime}, A^n, B^{\prime} A B$ are also symmetric matrix where $n \in N, k \in R$ and $B$ is a square matrix of order same as matrix $A$.
iii) If $A$ is a skew-symmetric matrix then
1. $A^{2 n}$ is a symmetric matrix for $n ? N$.
2. $A^{2 n+1}$ is a skew-symmetric matrix for $n$ ? $N$
3. $k A$ is also a skew-symmetric matrix, where $k \in R$
4. $B^{\prime} A B$ is also a skew-symmetric matrix where $B$ a square matrix of order same as matrix A
iv) If $A$ and $B$ are symmetric matrices then:
1. $A \pm B, A B+B A$ are symmetric matrices.
2. $A B-B A$ is a skew-symmetric matrix.
v) If A and B are skew-symmetric matrices then:
1. $A \pm B, A B-B A$ are skew-symmetric matrices.
2. $A B+B A$ is a symmetric matrix.