Question

write a square matrix of order 2 which is both symmetric and skew symmetric?

Answer 1

A= [0,0]
     [0,0]

and A' = [0,0]
              [0,0]
thus A=A'

also -A= [0,0]
              [0,0]
A=-A

Answer 2

The square matrix is $\bold{\left( \begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right)}$

Step-by-step explanation:

Method I:

Let us consider a matrix A of order 2 as $\left( \begin{array}{ll}{a} & {c} \\ {b} & {d}\end{array}\right)$ which is symmertric and skew symmetric

$A^{\top}=A \text { and } A^{\top}=(-A)$

$\left( \begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right)=\left( \begin{array}{ll}{a} & {c} \\ {b} & {d}\end{array}\right) \text { and } \left( \begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right)=\left( \begin{array}{cc}{-a} & {-c} \\ {-b} & {-d}\end{array}\right)$

a=a, b=c, c=b, d=d and a=-a, b=-b, c=-c, d=-d

b=c and 2a=0, 2b=0, 2c=0, 2d=0

a=b=c=d=0

The Matrix satisfying both the given conditions is a null matrix $\left( \begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right)$

Method II:

Given $\mathrm{A}^{\top}=\mathrm{A} \text { and } \mathrm{A}^{\mathrm{T}}=(-\mathrm{A})$

Adding both the equations, $A^{\top}+A^{\top}=A-A$

$2 \mathrm{A}^{\top}=0$

$A^{\top}=0$

$\mathrm{A}^{\mathrm{T}}=0$

Therefore, the matrix satisfying both the conditions is a null matrix $\left( \begin{array}{ll}{0} & {0} \\ {0} & {0}\end{array}\right)$