If A is a 3 × 3 matrix such that $|\text{adj A}| = 9$ and $|kA^{-1}|= 9$, then the value of k are:

$±3$

The correct answer is Option (3) → $±3$

Given:

• $| \text{adj} \, A | = 9$
• $| kA^{-1} | = 9$

For any invertible $3 \times 3$ matrix $A$:

$| \text{adj} \, A | = |A|^2$

So,

$|A|^2 = 9 \Rightarrow |A| = \pm 3$

Also, $|kA^{-1}| = |kI \cdot A^{-1}| = k^3 \cdot |A^{-1}|$

$|A^{-1}| = \frac{1}{|A|} = \frac{1}{\pm 3}$

So,

$k^3 \cdot \frac{1}{\pm 3} = 9$

$\Rightarrow k^3 = 9 \cdot (\pm 3) = \pm 27$

$\Rightarrow k = 3$ or $k = -3$