Question - Matrix Multiplication: Product of Matrices A and B
Solution:
The question is asking us to calculate the product of two matrices A and B:A = $$\begin{pmatrix} 3 & 1 \\ 5 & 4 \end{pmatrix}$$B = $$\begin{pmatrix} 5 & 2 & -1\\ 1 & 3 & 4 \end{pmatrix}$$Matrix A is a 2x2 matrix and matrix B is a 2x3 matrix. We can multiply A and B because the number of columns in A (which is 2) matches the number of rows in B (which is also 2). The resulting matrix will have the same number of rows as matrix A and the same number of columns as matrix B, which means the product will be a 2x3 matrix.Here's how we calculate the product AB:AB = $$\begin{pmatrix} 3 & 1 \\ 5 & 4 \end{pmatrix} \times \begin{pmatrix} 5 & 2 & -1\\ 1 & 3 & 4 \end{pmatrix}$$ AB = $$\begin{pmatrix} 3*5 + 1*1 & 3*2 + 1*3 & 3*(-1) + 1*4 \\5*5 + 4*1 & 5*2 + 4*3 & 5*(-1) + 4*4 \end{pmatrix}$$ AB = $$\begin{pmatrix} 15 + 1 & 6 + 3 & -3 + 4 \\25 + 4 & 10 + 12 & -5 + 16 \end{pmatrix}$$ AB = $$\begin{pmatrix} 16 & 9 & 1 \\29 & 22 & 11 \end{pmatrix}$$So the product of matrices A and B is the 2x3 matrix:$$\begin{pmatrix} 16 & 9 & 1 \\ 29 & 22 & 11 \end{pmatrix}$$
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