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Original Question
Find the product of the following two matrices 0 4 -1 -1 -3 0 0 -2Asked by d
Find the product of the following two matrices
3 -2| 1 -1
0 1| 1 1
3 -2| 1 -1
0 1| 1 1
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Answered by
GPT-4o mini
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:
1. Calculate \(C_{11}\):
\[
C_{11} = A_{11}B_{11} + A_{12}B_{21} = 3 \cdot 1 + (-2) \cdot 1 = 3 - 2 = 1
\]
2. Calculate \(C_{12}\):
\[
C_{12} = A_{11}B_{12} + A_{12}B_{22} = 3 \cdot (-1) + (-2) \cdot 1 = -3 - 2 = -5
\]
3. Calculate \(C_{21}\):
\[
C_{21} = A_{21}B_{11} + A_{22}B_{21} = 0 \cdot 1 + 1 \cdot 1 = 0 + 1 = 1
\]
4. 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:
\[
\begin{pmatrix}
1 & -5 \\
1 & 1
\end{pmatrix}
\]
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:
1. Calculate \(C_{11}\):
\[
C_{11} = A_{11}B_{11} + A_{12}B_{21} = 3 \cdot 1 + (-2) \cdot 1 = 3 - 2 = 1
\]
2. Calculate \(C_{12}\):
\[
C_{12} = A_{11}B_{12} + A_{12}B_{22} = 3 \cdot (-1) + (-2) \cdot 1 = -3 - 2 = -5
\]
3. Calculate \(C_{21}\):
\[
C_{21} = A_{21}B_{11} + A_{22}B_{21} = 0 \cdot 1 + 1 \cdot 1 = 0 + 1 = 1
\]
4. 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:
\[
\begin{pmatrix}
1 & -5 \\
1 & 1
\end{pmatrix}
\]
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