https://file.aiquickdraw.com/mathbot/file/1726643982735g741auv9.jpg

The image shows a \( 2 \times 2 \) matrix \( A \) and asks to find its inverse, \( A^{-1} \).

The matrix \( A \) is:
\[
A = \begin{pmatrix} 9 & 7 \\ 3 & 5 \end{pmatrix}
\]

The formula for the inverse of a \( 2 \times 2 \) matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is:
\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

For your matrix:
- \( a = 9 \)
- \( b = 7 \)
- \( c = 3 \)
- \( d = 5 \)

Let me calculate the determinant and the inverse matrix.The determinant of matrix \( A \) is \( 24 \), which means the matrix is invertible.

The inverse of matrix \( A \) is:
\[
A^{-1} = \begin{pmatrix} 0.2083 & -0.2917 \\ -0.125 & 0.375 \end{pmatrix}
\]

Would you like more details on the steps, or do you have any other questions?

Here are five related questions you might find useful:
1. How can I verify if a matrix is invertible without calculating the determinant?
2. What is the geometric interpretation of a matrix inverse?
3. How are inverse matrices used in solving systems of equations?
4. What happens when a matrix is singular (non-invertible)?
5. Can every square matrix be inverted?

**Tip:** Always check if the determinant of a matrix is non-zero before calculating its inverse!

Given a matrix A = [[9, 7], [3, 5]], find the inverse A^{-1}.

This solution explains how to find the inverse of a 2x2 matrix A = [[9, 7], [3, 5]] using the determinant formula and inverse matrix calculation method. The steps involve calculating the determinant and applying the inverse formula. Suitable for advanced high school students studying matrix algebra.

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