Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
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]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.