The matrix \( A = \begin{bmatrix} 2 & 2 \\ 3 & 5 \end{bmatrix} \) is a 2x2 matrix. To find the inverse of \( A \), we can use the formula for the inverse of a 2x2 matrix, which is:

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]
where the matrix is \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), and \( \text{det}(A) \) is the determinant of matrix \( A \), given by:

\[
\text{det}(A) = ad - bc
\]

For the matrix \( A = \begin{bmatrix} 2 & 2 \\ 3 & 5 \end{bmatrix} \):

- \( a = 2 \), \( b = 2 \), \( c = 3 \), and \( d = 5 \)
  
1. **Compute the determinant**:

\[
\text{det}(A) = (2)(5) - (2)(3) = 10 - 6 = 4
\]

2. **Apply the inverse formula**:

\[
A^{-1} = \frac{1}{4} \begin{bmatrix} 5 & -2 \\ -3 & 2 \end{bmatrix}
\]

3. **Simplify the matrix**:

\[
A^{-1} = \begin{bmatrix} \frac{5}{4} & -\frac{1}{2} \\ -\frac{3}{4} & \frac{1}{2} \end{bmatrix}
\]

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

\[
A^{-1} = \begin{bmatrix} \frac{5}{4} & -\frac{1}{2} \\ -\frac{3}{4} & \frac{1}{2} \end{bmatrix}
\]

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you verify if a matrix has an inverse?
2. What happens if the determinant of a matrix is zero?
3. How do you find the determinant of larger matrices (e.g., 3x3)?
4. Can all square matrices have inverses?
5. What are the properties of an inverse matrix?

**Tip**: Always check if the determinant is non-zero before finding the inverse of a matrix.

This page provides a detailed step-by-step solution on how to find the inverse of the 2x2 matrix A = [[2, 2], [3, 5]]. It includes explanations of the determinant formula and the process for finding the inverse using matrix algebra concepts. Suitable for high school students in Grades 10-12.

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