Question

Assertion :Every orthogonal matrix is invertible. Reason: If A is an orthogonal matrix then AA' is an identity matrix.

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

Let A be an orthogonal matrix As A is orthogonal matrix
$\therefore A{A}^T=I\Rightarrow \left|A{A}^T\right|=1$
$\Rightarrow \left|A\right|{\left||A\right|}^T=1$ $\Rightarrow \left|A^2\right|=1$ $\Rightarrow \left|A\right|=\pm 1$ $\Rightarrow$ A is non singular is $\Rightarrow A^{-1}$ exist $\Rightarrow A$ is invertible
$\therefore$ Both Assertion (A) & Reason (R) are true but Reason (R)
is not proper explanation of Assertion (A).