For a square matrix A of order 3, if \(|A| = 2\), then \(|\mathrm{adj}\, 2A| =\)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 256
For a square matrix B of order n, \(|\mathrm{adj}\,B| = |B|^{n-1}\). Here B = 2A and n = 3.
First, by \(|kA| = k^n|A|\):
\(|2A| = 2^3|A| = 8 \times 2 = 16\)
Then
\(|\mathrm{adj}\,2A| = |2A|^{3-1} = 16^2 = 256\)
Why the other options are incorrect
Option (1) 16 — Incorrect. This is \(|2A|\) itself; the determinant of the adjoint is \(|2A|^2\).
Option (2) 64 — Incorrect. \(|\mathrm{adj}\,2A| = |2A|^2 = 16^2 = 256\), not 64.
Option (3) 32 — Incorrect. This is \(|2\,\mathrm{adj}\,A| = 2^3 \times |A|^2 = 8 \times 4\). But \(\mathrm{adj}(2A) = 2^2\,\mathrm{adj}\,A\), not \(2\,\mathrm{adj}\,A\), since each cofactor of 2A is a 2 × 2 determinant and carries the factor \(2^2\).