To find the inverse of a 2x2 matrix, we can use the following formula. For a matrix
\[ A = \begin{pmatrix} a & b \ c & d \end{pmatrix}, \]
the inverse \( A^{-1} \) (if it exists) is given by:
\[ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}, \]
where \( ad - bc \) (the determinant of the matrix) must not be zero.
Given your matrix:
\[ A = \begin{pmatrix} 3 & 2 \ 5 & 3 \end{pmatrix}, \]
we identify \( a = 3 \), \( b = 2 \), \( c = 5 \), and \( d = 3 \).
Next, we calculate the determinant \( ad - bc \):
\[ \text{det}(A) = (3)(3) - (2)(5) = 9 - 10 = -1. \]
Since the determinant is not zero, the inverse exists. We can now substitute into the formula for the inverse:
\[ A^{-1} = \frac{1}{-1} \begin{pmatrix} 3 & -2 \ -5 & 3 \end{pmatrix} = \begin{pmatrix} -3 & 2 \ 5 & -3 \end{pmatrix}. \]
Thus, the inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} -3 & 2 \ 5 & -3 \end{pmatrix}. \]
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