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You're tasked with finding the inverse of the given \(2 \times 2\) matrix using an augmented matrix and elementary row operations. The matrix provided is:

\[
\begin{pmatrix}
-1 & 3 \\
2 & -5
\end{pmatrix}
\]

### Steps for finding the inverse using the augmented matrix:

1. **Set up the augmented matrix:**
   The augmented matrix combines the original matrix with the identity matrix. The goal is to perform row operations to turn the left side into the identity matrix, and the right side will become the inverse.
   
   \[
   \left[\begin{array}{cc|cc}
   -1 & 3 & 1 & 0 \\
   2 & -5 & 0 & 1
   \end{array}\right]
   \]

2. **Perform row operations:**
   - Step 1: Make the first entry in the first row a 1 (pivot at row 1, column 1).
     To achieve this, multiply row 1 by \(-1\).
     \[
     R_1 \rightarrow (-1) \cdot R_1
     \]
     Resulting matrix:
     \[
     \left[\begin{array}{cc|cc}
     1 & -3 & -1 & 0 \\
     2 & -5 & 0 & 1
     \end{array}\right]
     \]

   - Step 2: Eliminate the entry in row 2, column 1 (make it 0). To do this, subtract 2 times row 1 from row 2.
     \[
     R_2 \rightarrow R_2 - 2 \cdot R_1
     \]
     Resulting matrix:
     \[
     \left[\begin{array}{cc|cc}
     1 & -3 & -1 & 0 \\
     0 & 1 & 2 & 1
     \end{array}\right]
     \]

   - Step 3: Eliminate the entry in row 1, column 2 (make it 0). Add 3 times row 2 to row 1.
     \[
     R_1 \rightarrow R_1 + 3 \cdot R_2
     \]
     Resulting matrix:
     \[
     \left[\begin{array}{cc|cc}
     1 & 0 & 5 & 3 \\
     0 & 1 & 2 & 1
     \end{array}\right]
     \]

Now the left side of the matrix is the identity matrix, so the right side is the inverse matrix:

\[
\text{Inverse} =
\begin{pmatrix}
5 & 3 \\
2 & 1
\end{pmatrix}
\]

### Final Answer:
The inverse of the matrix is:

\[
\begin{pmatrix}
5 & 3 \\
2 & 1
\end{pmatrix}
\]

Do you need more details on any steps, or have any questions?

Here are 5 related questions to explore:
1. How do you know if a matrix is invertible?
2. What are elementary row operations?
3. Why does the augmented matrix method work for finding the inverse?
4. How can you verify the correctness of an inverse matrix?
5. What happens if the determinant of a matrix is 0?

**Tip:** The inverse of a matrix exists only if its determinant is non-zero! Always check the determinant first.

Write an augmented matrix and use elementary row operations in order to find the inverse of the matrix 
\begin{pmatrix} -1 & 3 \\ 2 & -5 \end{pmatrix} using reduced row echelon form.

Learn how to find the inverse of a 2x2 matrix using elementary row operations. This guided example explains the use of the augmented matrix method and row operations to achieve reduced row echelon form. Suitable for students in Grades 10-12.

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