Question

Wite a \[2 \times 2\] matrix which is both symmetric and skew symmetric.

Answer 1

Hint:A symmetric matrix and skew-symmetric matrix both are square matrices. But the difference between them is, the symmetric matrix is equal to its transpose where as skew-symmetric matrix is a matrix whose transpose is equal to its negative.If A is a symmetric matrix, then $A = {A^T}$ and if A is a skew-symmetric matrix then ${A^T} = - A$.Using these definitions we try to get the required answer.

Complete step-by-step answer:
We know that a $2 \times 2$ matrix contain $2$ row and $2$ column
As we know symmetric matrix means $A = {A^T}$
and skew-symmetric matrix then ${A^T} = - A$
where ${A^T}$ is transpose of matrix $A$
so as the given condition a matrix is both symmetric and skew symmetric.
That mean $A = - A$
So form this $A + A = 0$
$ \Rightarrow 2A = 0$, a null matrix
And $A = 0$, a null matrix
Hence, a $2 \times 2$ matrix which is both symmetric and skew symmetric is $\left[ {\begin{array}{*{20}{c}}
  0&0 \\
  0&0
\end{array}} \right]$
So our required answer is $\left[ {\begin{array}{*{20}{c}}
  0&0 \\
  0&0
\end{array}} \right]$

So, the correct answer is “Option A”.

Note:
Properties of Symmetric Matrix
1) Addition and difference of two symmetric matrices results in a symmetric matrix.
2) If $A$ and $B$ are two symmetric matrices and they follow the commutative property, i.e. $AB = BA$, then the product of $A$ and $B$ is symmetric.
3) If matrix $A$ is symmetric then ${A^n}$ is also symmetric, where n is an integer.
4) If $A$ is a symmetric matrix then ${A^{ - 1}}$ is also symmetric.

Properties of Skew Symmetric Matrix
1) When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric.
2) Scalar product of skew-symmetric matrices is also a skew-symmetric matrix.
3) The diagonal of skew symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero.
4) When the identity matrix is added to the skew symmetric matrix then the resultant matrix is invertible.
5) The determinant of skew symmetric matrix is non-negative