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Sagot :
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]
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]
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