If \(A = \begin{bmatrix} 2 & 5 & -7 \\ 0 & 3 & 4 \\ 0 & 0 & -1 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0 & 0 \\ 8 & 2 & 0 \\ -3 & 6 & 5 \end{bmatrix}\), then \(|AB|\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → −60
There is no need to multiply the matrices, because \(|AB| = |A|\,|B|\).
All the elements below the diagonal of A are 0, and all the elements above the diagonal of B are 0. Expanding such a triangular matrix along its first column (for A) or first row (for B) leaves only the product of the diagonal elements:
\(|A| = 2 \times 3 \times (-1) = -6\)
\(|B| = 1 \times 2 \times 5 = 10\)
By the product rule:
\(|AB| = (-6) \times 10 = -60\)