If \(M = \begin{bmatrix}2\\ -1\\ 3\end{bmatrix}\) and \(N = \begin{bmatrix}7 & 1 & -4\end{bmatrix}\), then \((MN)^T\) will be equal to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\begin{bmatrix}14 & -7 & 21\\ 2 & -1 & 3\\ -8 & 4 & -12\end{bmatrix}\)
M is 3 × 1 and N is 1 × 3, so MN is 3 × 3. Its (i, j) entry is the i-th entry of M times the j-th entry of N:
\( \displaystyle MN = \begin{bmatrix}2(7) & 2(1) & 2(-4)\\ -1(7) & -1(1) & -1(-4)\\ 3(7) & 3(1) & 3(-4)\end{bmatrix} \)
\( \displaystyle MN = \begin{bmatrix}14 & 2 & -8\\ -7 & -1 & 4\\ 21 & 3 & -12\end{bmatrix} \)
The transpose turns rows into columns:
\( \displaystyle (MN)^T = \begin{bmatrix}14 & -7 & 21\\ 2 & -1 & 3\\ -8 & 4 & -12\end{bmatrix} \)
Why the other options are incorrect
Option (1) \(\begin{bmatrix}7 & 1 & -4\\ 2 & -1 & 3\\ 9 & 0 & -1\end{bmatrix}\) — Incorrect. It is not the product at all: its first row is N, its second row is M written as a row, and its third row is their sum.
Option (2) \(\begin{bmatrix}14 & 2 & -8\\ -7 & -1 & 4\\ 21 & 3 & -12\end{bmatrix}\) — Incorrect. This is MN itself; the transpose has not been taken.
Option (3) \(\begin{bmatrix}14 & -7 & 21\\ 2 & -1 & 3\\ 1 & 3 & -12\end{bmatrix}\) — Incorrect. Its third row should be −8, 4, −12 (the third column of MN), not 1, 3, −12.