Question

If A is an orthogonal matrix & $\left|A\right|=-1,$ then $A^T$ is equal to -

• A. $-A$
• B. $A$
• C. $-\left(adjA\right)$
• D. $\left(adjA\right)$

Answer 1

The correct option is A $-A$

In an orthogonal matrix $detA=\pm 1$

here given that $\left|A\right|=-1$

$A{A}^T=I$

$\left|A{A}^T\right|=1\left|A\right|.\left|A^T\right|=1-1.\left|A^T\right|=1\left|A^T\right|=-1A^T=-A\left|A^T\right|=-\left|A\right|=-(-1)=1\therefore A^T=-A$