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You are asked to find the inverse of the matrix \( A \):

\[
A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}
\]

To find the inverse of a 2x2 matrix, we use the formula:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]

Where the matrix \( A \) is:

\[
A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}
\]

For the given matrix:

\[
A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}, \quad a = 2, b = 3, c = 3, d = 5
\]

### Step 1: Compute the determinant of \( A \):

\[
\text{det}(A) = ad - bc = (2)(5) - (3)(3) = 10 - 9 = 1
\]

### Step 2: Find the inverse matrix:

Using the formula, the inverse is:

\[
A^{-1} = \frac{1}{1} \begin{bmatrix} 5 & -3 \\ -3 & 2 \end{bmatrix}
\]

Thus, the inverse of matrix \( A \) is:

\[
A^{-1} = \begin{bmatrix} 5 & -3 \\ -3 & 2 \end{bmatrix}
\]

Let me know if you'd like further details, or have any questions. Here are some related questions you might find helpful:

1. What are the properties of the inverse of a matrix?
2. Can a matrix always have an inverse?
3. How do you find the inverse of larger matrices (e.g., 3x3 matrices)?
4. What is the determinant of a matrix and its significance?
5. How is the inverse of a matrix used in solving systems of linear equations?

**Tip:** The inverse of a matrix exists only if its determinant is non-zero. If the determinant is zero, the matrix is called singular, and no inverse exists.

Find the inverse of the matrix A = [[2, 3], [3, 5]].

Learn how to find the inverse of a 2x2 matrix A = [[2, 3], [3, 5]] using the determinant and matrix inversion formula. A step-by-step guide suitable for high school students studying linear algebra.

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