If $A=\begin{bmatrix}-2\\-1\\-4\end{bmatrix}, B = \begin{bmatrix}-1&2&3\end{bmatrix}$, then the value of $A' B'$ is

$\begin{bmatrix}-12\end{bmatrix}$

The correct answer is Option (4) → $\begin{bmatrix}-12\end{bmatrix}$

Given:

$A = \begin{bmatrix} -2 \\ -1 \\ -4 \end{bmatrix}$, $B = \begin{bmatrix} -1 & 2 & 3 \end{bmatrix}$

$A' = \text{transpose of } A = \begin{bmatrix} -2 & -1 & -4 \end{bmatrix}$
$B' = \text{transpose of } B = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$

Then:

$A'B' = \begin{bmatrix} -2 & -1 & -4 \end{bmatrix} \cdot \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$

$=[ (-2)(-1) + (-1)(2) + (-4)(3) ]= [2 - 2 - 12] = [-12]$