To determine if the two matrices are inverses of each other, we need to multiply them together. If the product of the two matrices is the identity matrix, then they are inverses.
Let \( A = \begin{pmatrix} 1 & 2 \ 3 & 5 \end{pmatrix} \) and \( B = \begin{pmatrix} 5 & -2 \ -3 & 1 \end{pmatrix} \).
Now, we will calculate the product \( A \times B \):
\[ A \times B = \begin{pmatrix} 1 & 2 \ 3 & 5 \end{pmatrix} \begin{pmatrix} 5 & -2 \ -3 & 1 \end{pmatrix} \]
To find the resulting matrix, we perform the matrix multiplication:
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The element in the first row, first column: \[ 1 \cdot 5 + 2 \cdot (-3) = 5 - 6 = -1 \]
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The element in the first row, second column: \[ 1 \cdot (-2) + 2 \cdot 1 = -2 + 2 = 0 \]
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The element in the second row, first column: \[ 3 \cdot 5 + 5 \cdot (-3) = 15 - 15 = 0 \]
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The element in the second row, second column: \[ 3 \cdot (-2) + 5 \cdot 1 = -6 + 5 = -1 \]
Putting it all together, we get:
\[ A \times B = \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \]
Next, we check if this product is the identity matrix:
The identity matrix for a \(2 \times 2\) matrix is:
\[ I = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]
Since \( A \times B = \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} \neq I \), the two matrices are not inverses of each other.
Additionally, note that \( A \times B \) is actually \(-I\). This means \( B \) is the inverse of \( A \) up to a scalar factor of \(-1\) in the context of \(GL(2, \mathbb{R})\), but strictly, they are not inverses as defined.
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