To determine the product of the matrices [tex]\(\left[\begin{array}{lll}0 & -1 & -4\end{array}\right]\)[/tex] and [tex]\(\left[\begin{array}{c}-2 \\ 5 \\ -1\end{array}\right]\)[/tex], we follow the steps for matrix multiplication.
Given:
Matrix [tex]\( A = \left[ \begin{array}{ccc} 0 & -1 & -4 \end{array} \right] \)[/tex]
Matrix [tex]\( B = \left[ \begin{array}{c} -2 \\ 5 \\ -1 \end{array} \right] \)[/tex]
The product of these matrices can be computed as:
[tex]\[
A \cdot B = \left[ \begin{array}{ccc} 0 & -1 & -4 \end{array} \right] \cdot \left[ \begin{array}{c} -2 \\ 5 \\ -1 \end{array} \right]
\][/tex]
To multiply them, you take the dot product of the row vector from matrix [tex]\( A \)[/tex] with the column vector from matrix [tex]\( B \)[/tex]:
[tex]\[
\left( 0 \times (-2) \right) + \left( -1 \times 5 \right) + \left( -4 \times (-1) \right)
\][/tex]
Calculating each term:
[tex]\[
0 \cdot (-2) = 0
\][/tex]
[tex]\[
-1 \cdot 5 = -5
\][/tex]
[tex]\[
-4 \cdot (-1) = 4
\][/tex]
Now, add these results together:
[tex]\[
0 + (-5) + 4 = -1
\][/tex]
Therefore, the product of the matrices is:
[tex]\[
-1
\][/tex]
So the answer is [tex]\(-1\)[/tex].
