https://file.aiquickdraw.com/mathbot/file/1729387521952pksc2ef8.jpg

The matrix provided is:

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

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

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

Where:
- \(\text{det}(A)\) is the determinant of the matrix \(A\).
- \(\text{adj}(A)\) is the adjugate (or adjoint) of the matrix \(A\), which is the transpose of its cofactor matrix.

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

For a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the determinant is:

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

Substituting the values from matrix \(A\):

\[
\text{det}(A) = (2)(-2) - (-3)(-1) = -4 - 3 = -7
\]

### Step 2: Find the adjugate matrix

The adjugate of a 2x2 matrix is:

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

For matrix \(A = \begin{pmatrix} 2 & -3 \\ -1 & -2 \end{pmatrix}\), this becomes:

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

### Step 3: Calculate the inverse

Now, we multiply the adjugate matrix by \(\frac{1}{\text{det}(A)}\):

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

Distribute \(\frac{1}{-7}\) across the matrix:

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

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

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

Do you want more details, or do you have any questions?

### Related Questions:
1. How do you calculate the determinant of a 3x3 matrix?
2. What are the conditions for a matrix to have no inverse?
3. Can a matrix with a zero determinant have an inverse?
4. How do you use matrix inverses to solve systems of equations?
5. What is the adjugate matrix, and how is it used in larger matrices?

### Tip:
To quickly check if a 2x2 matrix is invertible, look at its determinant. If it's nonzero, the matrix has an inverse!

Find the inverse of the following matrix, if possible. Reduce all fractions to lowest terms. A = [[2, -3], [-1, -2]]

Learn how to find the inverse of a 2x2 matrix A = [[2, -3], [-1, -2]]. This step-by-step guide covers matrix inversion using determinants, adjugates, and fraction reduction. Suitable for high school and early university students.

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