Question

Assertion : If $A$ is a non-singular matrix, then $|A^{-}1|=\frac{1}{|A|}$
Reason : If $A$ and $B$ are inverse matrices of each other , then $AB=BA=I$ and $|AB|=|A||B|$

• A. Assertion and reason both are true and reason is the correct explanation of assertion.
• B. Assertion and reason both are true but reason is not the correct explanation of assertion.
• C. Assertion is false
• D. Reason is false

Answer 1

Answer: (C)

Assertion is false

If $A$ is a non-singular matrix, then

$A^{-1}=\frac{1}{\left|A\right|}adjA$

Hence, assertion is wrong.

$=\frac{(ax-by+cz)}{a^2}\left|\begin{array}{ccc}ax-by+cz& ac-ac& ab-ab\\ y& 0& a\\ z& -a& 0\end{array}\right|$

$\therefore R_1\to R_1-b{R}_2+c{R}_3$

$=\frac{(ax-by+cz)}{a^2}\left|\begin{array}{ccc}ax-by+cz& 0& 0\\ y& 0& a\\ z& -a& 0\end{array}\right|$

$=\frac{(ax-by+cz)}{a^2}(ax-by+cz)\left(a^2\right)$

$={(ax-by+cz)}^2$

Hence, the answer is assertion is false.