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The problem asks to show that matrices \( A \) and \( B \) are invertible and that they are inverses of each other. The given matrices are:

\[
A = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix}
\]

To show that \( A \) and \( B \) are inverses of each other, we need to verify that:

\[
A \cdot B = B \cdot A = I
\]

Where \( I \) is the identity matrix:

\[
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

### Step 1: Compute \( A \cdot B \)

\[
A \cdot B = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \cdot \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix}
\]

The multiplication is done as follows:

\[
A \cdot B = \begin{pmatrix} (2 \cdot 3 + 5 \cdot (-1)) & (2 \cdot (-5) + 5 \cdot 2) \\ (1 \cdot 3 + 3 \cdot (-1)) & (1 \cdot (-5) + 3 \cdot 2) \end{pmatrix}
\]

Simplifying each element:

\[
A \cdot B = \begin{pmatrix} (6 - 5) & (-10 + 10) \\ (3 - 3) & (-5 + 6) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

So, \( A \cdot B = I \).

### Step 2: Compute \( B \cdot A \)

\[
B \cdot A = \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix} \cdot \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}
\]

Similarly, the multiplication is:

\[
B \cdot A = \begin{pmatrix} (3 \cdot 2 + (-5) \cdot 1) & (3 \cdot 5 + (-5) \cdot 3) \\ (-1 \cdot 2 + 2 \cdot 1) & (-1 \cdot 5 + 2 \cdot 3) \end{pmatrix}
\]

Simplifying each element:

\[
B \cdot A = \begin{pmatrix} (6 - 5) & (15 - 15) \\ (-2 + 2) & (-5 + 6) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

So, \( B \cdot A = I \).

### Conclusion

Since \( A \cdot B = I \) and \( B \cdot A = I \), we have shown that \( A \) and \( B \) are inverses of each other. Therefore, both \( A \) and \( B \) are invertible matrices, and they are each other's inverses.

Would you like more details or explanations on any part of this? 

Here are 5 related questions for further practice:

1. How do you determine if a matrix is invertible?
2. What is the general formula for the inverse of a 2x2 matrix?
3. Can a non-square matrix be invertible? Why or why not?
4. How does matrix multiplication differ from scalar multiplication?
5. Why is the identity matrix crucial in matrix operations?

**Tip:** The inverse of a matrix \( A \), when it exists, satisfies \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix.

Suppose A = [[2, 5], [1, 3]] and B = [[3, -5], [-1, 2]]. Then show that A and B are invertible and are inverses of each other.

This problem demonstrates how to check if two 2x2 matrices are invertible and if they are inverses of each other. Learn matrix multiplication techniques and verify the identity matrix I for Grades 10-12 or undergraduate students.

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