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To find the inverse of the matrix \( A \), where 

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

we can follow these steps:

### 1. Compute the Determinant of \( A \):
The determinant of a \( 2 \times 2 \) matrix 

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

is given by:

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

For matrix \( A \):

\[
\text{det}(A) = (3)(2) - (1)(-1) = 6 + 1 = 7
\]

### 2. Find the Adjoint of \( A \):
The adjoint (or adjugate) of a \( 2 \times 2 \) matrix \( A \) is formed by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements:

\[
\text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]

So, for matrix \( A \):

\[
\text{adj}(A) = \begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix}
\]

### 3. Compute the Inverse of \( A \):
The inverse of matrix \( A \) is given by:

\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]

Substituting the values we have:

\[
A^{-1} = \frac{1}{7} \cdot \begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix}
\]

This simplifies to:

\[
A^{-1} = \begin{bmatrix} \frac{2}{7} & -\frac{1}{7} \\ \frac{1}{7} & \frac{3}{7} \end{bmatrix}
\]

### Final Answer:
The inverse of the matrix \( A \) is:

\[
A^{-1} = \begin{bmatrix} \frac{2}{7} & -\frac{1}{7} \\ \frac{1}{7} & \frac{3}{7} \end{bmatrix}
\]

Would you like any additional details or have further questions?

Here are some related questions you might be interested in:

1. How do you find the determinant of a 3x3 matrix?
2. What are the properties of the inverse of a matrix?
3. How can you verify that a matrix is invertible?
4. What happens if the determinant of a matrix is zero?
5. Can you find the inverse of a matrix using row operations?

**Tip:** Always verify the invertibility of a matrix by checking if its determinant is non-zero before attempting to find its inverse.

Learn how to find the inverse of matrix A = [3 1; -1 2] in fraction form step-by-step. This problem covers key concepts in matrix algebra, including determinants, adjoints, and matrix inversion. Suitable for undergraduate students.

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