To find the product of the given matrix and the vector, we shall go through the matrix multiplication step-by-step.
Given:
[tex]\[ A = \begin{pmatrix}
0 & 0 & 1 \\
2 & 4 & 0 \\
0 & 0 & 2
\end{pmatrix}, \][/tex]
and
[tex]\[ B = \begin{pmatrix}
2 \\
0 \\
1
\end{pmatrix}, \][/tex]
we are to compute [tex]\( AB \)[/tex].
Matrix multiplication is performed by taking the dot product of each row of the first matrix with the column vector.
Let's perform the computations:
1. First Row Dot Product:
[tex]\[
\begin{pmatrix}
0 & 0 & 1
\end{pmatrix}
\cdot
\begin{pmatrix}
2 \\
0 \\
1
\end{pmatrix}
\][/tex]
Calculation:
[tex]\[
0 \cdot 2 + 0 \cdot 0 + 1 \cdot 1 = 0 + 0 + 1 = 1
\][/tex]
2. Second Row Dot Product:
[tex]\[
\begin{pmatrix}
2 & 4 & 0
\end{pmatrix}
\cdot
\begin{pmatrix}
2 \\
0 \\
1
\end{pmatrix}
\][/tex]
Calculation:
[tex]\[
2 \cdot 2 + 4 \cdot 0 + 0 \cdot 1 = 4 + 0 + 0 = 4
\][/tex]
3. Third Row Dot Product:
[tex]\[
\begin{pmatrix}
0 & 0 & 2
\end{pmatrix}
\cdot
\begin{pmatrix}
2 \\
0 \\
1
\end{pmatrix}
\][/tex]
Calculation:
[tex]\[
0 \cdot 2 + 0 \cdot 0 + 2 \cdot 1 = 0 + 0 + 2 = 2
\][/tex]
Therefore, the resulting product [tex]\( AB \)[/tex] is:
[tex]\[ AB = \begin{pmatrix}
1 \\
4 \\
2
\end{pmatrix}. \][/tex]
Hence, the result of the matrix-vector multiplication is:
[tex]\[
\begin{pmatrix}
1 \\
4 \\
2
\end{pmatrix}.
\][/tex]
