If A is a square matrix of order 3 such that \(|A| = -3\), then the value of \(|AA^{T}|\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 9
By the product rule, \(|AB| = |A|\,|B|\) for square matrices of the same order. Also a matrix and its transpose have the same determinant, so \(|A^{T}| = |A| = -3\).
\( |AA^{T}| = |A| \times |A^{T}| \)
\( = (-3) \times (-3) \)
\( = 9 \)
Why the other options are incorrect
Option (1) \(-9\) — Incorrect. This takes \(|A^{T}| = 3\); transposing does not change the sign of the determinant.
Option (3) 6 — Incorrect. The two determinants are multiplied, not added: \((-3) \times (-3) = 9\).
Option (4) \(-6\) — Incorrect. This adds the determinants; the product rule multiplies them.