If \(A = \begin{bmatrix}2&0&0\\0&-1&0\\0&0&3\end{bmatrix}\) and \(B = \begin{bmatrix}1&0&0\\0&4&0\\0&0&-2\end{bmatrix}\), then the sum of the diagonal elements of AB is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(-8\)
The product of two diagonal matrices of the same order is a diagonal matrix whose diagonal elements are the products of the corresponding diagonal elements. For example, row 1 of A times column 1 of B is \(2(1) + 0 + 0 = 2\). So
\( \displaystyle AB = \begin{bmatrix}2&0&0\\0&-4&0\\0&0&-6\end{bmatrix} \)
The sum of its diagonal elements is
\(2 - 4 - 6 = -8\)
Why the other options are incorrect
Option (1) 12 — Incorrect. This multiplies the diagonal sums 4 and 3 of A and B; the diagonal sum of AB is not that product.
Option (3) 7 — Incorrect. This is the diagonal sum of \(A + B\), not of AB.
Option (4) \(-6\) — Incorrect. This is only the last diagonal element of AB.