Question - Matrix Multiplication Example
Solution:
The image shows two matrices that we are supposed to multiply. Let's call the first matrix A and the second matrix B:Matrix A:\[ \begin{bmatrix} 3 & 2 & 5 \\ 2 & 3 & 1 \end{bmatrix} \]Matrix B:\[ \begin{bmatrix} 4 & 5 & -5 \\ 5 & -1 & 6 \\ 1 & 6 & -1 \end{bmatrix} \]To multiply these matrices, each element of the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the corresponding column from the second matrix.The resulting matrix will have the same number of rows as the first matrix (2 rows) and the same number of columns as the second matrix (3 columns).Let's calculate the elements of the resulting matrix one by one:Top-left element (C11) = (3 * 4) + (2 * 5) + (5 * 1) = 12 + 10 + 5 = 27Top-middle element (C12) = (3 * 5) + (2 * -1) + (5 * 6) = 15 - 2 + 30 = 43Top-right element (C13) = (3 * -5) + (2 * 6) + (5 * -1) = -15 + 12 - 5 = -8Bottom-left element (C21) = (2 * 4) + (3 * 5) + (1 * 1) = 8 + 15 + 1 = 24Bottom-middle element (C22) = (2 * 5) + (3 * -1) + (1 * 6) = 10 - 3 + 6 = 13Bottom-right element (C23) = (2 * -5) + (3 * 6) + (1 * -1) = -10 + 18 - 1 = 7Now we can construct the resulting matrix C:\[ \begin{bmatrix}27 & 43 & -8 \\24 & 13 & 7\end{bmatrix} \]So, the product of the two matrices A and B is:\[ \begin{bmatrix}27 & 43 & -8 \\24 & 13 & 7\end{bmatrix} \]
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