If \(A = \begin{bmatrix}2 & -1\\ 3 & 4\end{bmatrix}\) and \(B = \begin{bmatrix}1 & 2\\ -3 & 1\end{bmatrix}\), then the value of \(|AB|\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 77
By the product rule, \(|AB| = |A|\,|B|\), so the product AB need not be found.
\(|A| = 2 \cdot 4 - (-1) \cdot 3 = 11\)
\(|B| = 1 \cdot 1 - 2 \cdot (-3) = 7\)
\(|AB| = 11 \times 7 = 77\)
Check: \(AB = \begin{bmatrix}5 & 3\\ -9 & 10\end{bmatrix}\), and \(50 - (-27) = 77\).
Why the other options are incorrect
Option (2) 18 — Incorrect. This is \(|A| + |B|\); the determinants multiply, they do not add.
Option (3) \(-77\) — Incorrect. Both \(|A| = 11\) and \(|B| = 7\) are positive, so the product is positive.
Option (4) 4 — Incorrect. This is \(|A| - |B|\), not the product.