Question

Assertion :If A is a non-singular symmetric matrix, then its inverse is also symmetric.
$Because$ Reason: $(A^{-1}{)}^T=(A^T{)}^{-1}$, where A is a non-singular symmetric 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $A$ be an invertible matrix. Than there exists a matrix $B$ such that

$AB=I_n=BA$

$\Rightarrow \left|AB\right|=\left|I_n\right|$

$\Rightarrow \left|A\right|\left|B\right|=1(\because \left|AB\right|=\left|A\right|\left|B\right|)$

$\Rightarrow \left|A\right|\ne 0$

$\Rightarrow A$ is non-singular matrix.

Conversely, if A is non-singular, then

$A.\frac{\left(adjA\right)}{\left(A\right)}=I_n\Rightarrow A^{-1}=\frac{adjA}{\left|A\right|}$