For a square matrix A of order 3, if $|A| = 2$, then $|adj\, 2A| =$

256

The correct answer is Option (4) → 256

Given: $|A| = 2,\; A$ is a $3\times3$ matrix.

For any square matrix of order $n$, $|\text{adj}A| = |A|^{n-1}$

Thus, $|\text{adj}A| = 2^{3-1} = 2^{2} = 4$

Now for $2A$:

$|\text{adj}(2A)| = |2A|^{n-1} = (2^{n}|A|)^{n-1}$

For $n = 3$:

$|\text{adj}(2A)| = (2^{3}\cdot2)^{2} = (16)^{2} = 256$

$|\text{adj}(2A)| = 256$