If \(A =\begin{bmatrix}a&1&-1\\0&b&4\\4&4&c\end{bmatrix}\) and \(abc = 12, b = 4a\), then the value of \(|A(\mathrm{adj}A)|\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(28^3\)
For a square matrix of order 3, \(A(\mathrm{adj}\,A) = |A|\,I\), so
\(|A(\mathrm{adj}\,A)| = \Big||A|\,I\Big| = |A|^3\)
Expanding |A| along the first row:
\(|A| = a(bc - 16) - 1(0 - 16) + (-1)(0 - 4b)\)
\(= abc - 16a + 16 + 4b\)
Putting \(abc = 12\) and \(b = 4a\):
\(|A| = 12 - 16a + 16 + 16a = 28\)
Hence
\(|A(\mathrm{adj}\,A)| = 28^3\)
Why the other options are incorrect
Option (2) \(12^3\) — Incorrect. 12 is the value of abc, not of |A|; the terms −16a + 16 + 4b also add to |A|.
Option (3) \(28^2\) — Incorrect. \(\Big||A|\,I\Big| = |A|^n\) with n = 3, the order of the matrix, so the power is 3, not 2.
Option (4) \(48^2\) — Incorrect. \(48^2 = 2304\), but \(|A(\mathrm{adj}\,A)| = 28^3 = 21952\).