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To solve this problem, we need to find the inverse of the given matrix \( A = \begin{bmatrix} 1 & -1 \\ 5 & -2 \end{bmatrix} \) using the provided formula.

The inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:

\[
A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]

For the given matrix \( A = \begin{bmatrix} 1 & -1 \\ 5 & -2 \end{bmatrix} \), the values of \( a \), \( b \), \( c \), and \( d \) are:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 5 \)
- \( d = -2 \)

### Step 1: Calculate the determinant
The determinant of matrix \( A \) is:

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

### Step 2: Apply the formula for the inverse
Using the formula for the inverse:

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

This simplifies to:

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

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

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

Would you like to see further details or explanations about this? Here are some questions to expand on this topic:

1. How do you check if a matrix is invertible?
2. What happens if the determinant is zero?
3. How do you find the determinant of larger matrices?
4. Can you explain matrix multiplication and its properties?
5. What are some real-life applications of matrix inversion?

**Tip:** Always check the determinant before calculating the inverse, as a zero determinant means the matrix is non-invertible.

Use the formula for finding the inverse of a 2x2 matrix A = [1 -1; 5 -2] to determine if it has an inverse, and if so, calculate the inverse.

Learn how to determine if the matrix A = [1 -1; 5 -2] has an inverse. The solution involves calculating the determinant and applying the inverse matrix formula for a 2x2 matrix. Suitable for high school students in Grades 10-12.

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