If A = […] , then the value of det (adj (2A)) is:

The correct answer is Option (4) → 1600

For a square matrix M of order n, \(|kM| = k^n|M|\) and \(|\mathrm{adj}\,M| = |M|^{n-1}\). Here n = 3.

Expanding |A| along the first column, which has a zero:

    \(\displaystyle |A| = 2\begin{vmatrix}2&-1\\-5&0\end{vmatrix} + 3\begin{vmatrix}-1&-2\\2&-1\end{vmatrix}\)

    \(= 2(0 - 5) + 3(1 + 4) = 5\)

Then

    \(|2A| = 2^3|A| = 8 \times 5 = 40\)
    \(|\mathrm{adj}(2A)| = |2A|^{3-1} = 40^2 = 1600\)

Tip: For a 3 × 3 matrix, \(|2A| = 2^3|A|\), not 2|A|. Taking |2A| = 10 gives 10² = 100, Option (1).