Question

If $A$ is a $3 \times 3$ nonsingular matrix and if $|A | = 3$, then $|(2A)^{-1}|$= _______

• A. $\frac {1}{3}$
• B. $\frac {1}{24}$
• C. $24$
• D. $3$

Answer 1

Answer: (B)

$\frac {1}{24}$

Answer 2

Given, $|A|_{3 \times 3} \neq 0$ and $|A|=3$
Then, $\left|(2 A)^{-1}\right|=\left|\frac{1}{2 A}\right|=\frac{1}{|2 A|}$
$=\frac{1}{(2)^{3}} \cdot \frac{1}{|A|} \left(\because|a A|=a^{3}|A|\right)$
$=\frac{1}{8} \cdot \frac{1}{3}=\frac{1}{24}$