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 ${\mathrm{A}}^{\prime }=\mathrm{A}$,

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

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

Skew-symmetric matrix:  

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

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

$\begin{array}{rl}& {\mathrm{a}}_{\mathrm{ii}}=-{\mathrm{a}}_{\mathrm{ii}}\\ & \therefore 2{\mathrm{a}}_{\mathrm{ii}}=0\Rightarrow {\mathrm{a}}_{\mathrm{ii}}=0\mathrm{\forall }{\mathrm{i}}^{\prime }\mathrm{s}\end{array}$
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 $-A,kA,A^{\prime },A^n,B^{\prime }AB$ 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. $A^{2n}$ is a symmetric matrix for $n?N$.
2. $A^{2n+1}$ is a skew-symmetric matrix for $n$ ? $N$
3. $kA$ is also a skew-symmetric matrix, where $k\in R$
4. $B^{\prime }AB$ 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. $A\pm B,AB+BA$ are symmetric matrices.
2. $AB-BA$ is a skew-symmetric matrix.
v) If A and B are skew-symmetric matrices then:
1. $A\pm B,AB-BA$ are skew-symmetric matrices.
2. $AB+BA$ is a symmetric matrix.