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)

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