To find the product of two matrices, we follow the matrix multiplication rules. Let’s denote the matrices as follows:
Matrix A: \[ A = \begin{pmatrix} 3 & -2 \ 0 & 1 \end{pmatrix} \]
Matrix B: \[ B = \begin{pmatrix} 1 & -1 \ 1 & 1 \end{pmatrix} \]
The product \(C = AB\) is calculated using the formula for matrix multiplication:
\[ C_{ij} = \sum_{k} A_{ik} B_{kj} \]
Now, we will multiply matrix A by matrix B:
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Calculate \(C_{11}\): \[ C_{11} = A_{11}B_{11} + A_{12}B_{21} = 3 \cdot 1 + (-2) \cdot 1 = 3 - 2 = 1 \]
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Calculate \(C_{12}\): \[ C_{12} = A_{11}B_{12} + A_{12}B_{22} = 3 \cdot (-1) + (-2) \cdot 1 = -3 - 2 = -5 \]
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Calculate \(C_{21}\): \[ C_{21} = A_{21}B_{11} + A_{22}B_{21} = 0 \cdot 1 + 1 \cdot 1 = 0 + 1 = 1 \]
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Calculate \(C_{22}\): \[ C_{22} = A_{21}B_{12} + A_{22}B_{22} = 0 \cdot (-1) + 1 \cdot 1 = 0 + 1 = 1 \]
Putting it all together, we have: \[ C = \begin{pmatrix} C_{11} & C_{12} \ C_{21} & C_{22} \end{pmatrix} = \begin{pmatrix} 1 & -5 \ 1 & 1 \end{pmatrix} \]
Thus, the product of the two matrices is: \[ \begin{pmatrix} 1 & -5 \ 1 & 1 \end{pmatrix} \]