Question

Assertion :Suppose $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$
Let $X$ be a $2\times 2$ matrices such that $X^{\prime }AX=B$.

$X$ is non-singular and $det\left(X\right)=\pm 2$

Reason: $X$ is a singular matrix

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

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Let $det\left(X\right)=a$, then $det\left(X^{\prime }\right)=a$
$\therefore det\left(X^{\prime }AX\right)=det\left(B\right)$
$\Rightarrow a(-1)a=-4\Rightarrow a=\pm 2$
As $det\left(X\right)\ne 0$, $X$ cannot be a singular matrix.
Assertion is correct but Reason is incorrect
Hence, option C.