If A is a square matrix of order 3 and \(|A| = -4\), then the value of \(|3A^{T}|\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(-108\)
Two facts are used:
\(|A^{T}| = |A|\)
\(|kB| = k^{n}|B|\) for a matrix B of order n
In 3A every one of the 3 rows is multiplied by 3, so a factor 3 comes out of each row:
\(|3A^{T}| = 3^{3}|A^{T}|\)
\(|3A^{T}| = 27 \times (-4)\)
\(|3A^{T}| = -108\)
Why the other options are incorrect
Option (2) \(-36\) — Incorrect. This uses \(3^{2}\), the factor for a matrix of order 2.
Option (3) 108 — Incorrect. Transposing does not change the sign: \(|A^{T}| = |A| = -4\).
Option (4) \(-12\) — Incorrect. This takes the factor 3 only once; it comes out of each of the 3 rows.