Question

Assertion :Inverse of an orthogonal matrix is orthogonal. Reason: The inverse of an identity matrix is the matrix itself.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R} is false,
• D. (A)is false but (R} is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),

Let $A$ be an orthogonal matrix.
$\therefore A{A}^{\prime }=I$

$\Rightarrow (A{A}^{\prime }{)}^{-1}=I$
$\Rightarrow (A^{\prime }{)}^{-1}{A}^{-1}=I$

$\Rightarrow \left(A^{-1}{)}^{\prime }\right(A^{-1})=I$

$\Rightarrow A^{-1}$ is orthogonal
$\therefore$ Assertion (A) and Reason (R) both are individually true and Reason (R) is correct explanation of Assertion (A).