Question

Assertion :Let A be a square matrix given by $A=\left(\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right)$ then skew symmetric part of A is given by $\left(\begin{array}{ccc}0& -2& 1\\ 2& 0& 2\\ -1& -2& 0\end{array}\right)$ Reason: For every square matrix A -A' is skew Symmetric.

• A. Both Assertion and Reason are correct and the Reason is the correct explanation for the Assertion.
• B. Both Assertion and Reason are correct but the Reason is Not the correct explanation for the Assertion.
• C. Assertion is correct while the Reason is incorrect.
• D. Assertion is incorrect while the Reason is correct.

Answer 1

The correct option is B Both Assertion and Reason are correct but the Reason is Not the correct explanation for the Assertion.

Let $P=A-A^{\prime }$
$\Rightarrow P^{\prime }=(A-A^{\prime }{)}^{\prime }=A^{\prime }-(A^{\prime }{)}^{\prime }=-(A-A^{\prime })=-P\Rightarrow A-A^{\prime }$ is skew symmetric.
Now $A=\left[\begin{array}{ccc}1& 2& 4\\ 6& 8& 2\\ 2& -2& 7\end{array}\right]$
$\Rightarrow A^{\prime }=\left[\begin{array}{ccc}1& 6& 2\\ 2& 8& -2\\ 4& 2& 7\end{array}\right]$
$\therefore A-A^{\prime }=\left[\begin{array}{ccc}0& -4& 2\\ 4& 0& 4\\ -2& -4& 0\end{array}\right]$
$\Rightarrow \frac{1}{2}(A-A^{\prime })=\left[\begin{array}{ccc}0& -2& 1\\ 2& 0& 2\\ -1& -2& 0\end{array}\right]$ which is skew Symmetric matrix.
$\therefore$ Both Assertion and Reason are individually true but the Reason is not the correct explanation of the Assertion.