Question

If $A$ is a $3 \times 3$ non-singular matrix, then $\mid A^{-1}$ adj $A \mid$ is

• A. $|A|$
• B. 1
• C. $|A|^2$
• D. $|A|^{-1}$

Answer 1

Answer: (A)

$\mid A^{-1} \text { adj } A \mid =\left|A^{-1}\right| \cdot \text { adj } A \mid (\because|A B|=|A| \cdot|B|) $
$ =|A|^{-1} \mid \text { adj } A \mid \left(\because\left|A^{-1}\right|=|A|^{-1}\right) $
$ =|A|^{-1} \cdot|A|^2 $
$ (\because A \text { is a } 3 \times 3 $ non-singular matrix, so$ \mid \text { adj }\left.A|=| A\right|^2) $
$ =|A|$