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The problem asks us to verify that the product of two matrices \( A \) and \( B \) results in the identity matrix, i.e., \( AB = BA = I \), where:

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

We will first compute the matrix product \( AB \), then compute \( BA \), and check if both products result in the identity matrix \( I \).

### Step 1: Compute \( AB \)

\[
AB = \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix} \times \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix}
\]

Multiplying the two matrices:
- First row, first column: \( (2 \times 3) + (5 \times -1) = 6 - 5 = 1 \)
- First row, second column: \( (2 \times -5) + (5 \times 2) = -10 + 10 = 0 \)
- Second row, first column: \( (1 \times 3) + (3 \times -1) = 3 - 3 = 0 \)
- Second row, second column: \( (1 \times -5) + (3 \times 2) = -5 + 6 = 1 \)

Thus,

\[
AB = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

This is the identity matrix \( I \).

### Step 2: Compute \( BA \)

\[
BA = \begin{bmatrix} 3 & -5 \\ -1 & 2 \end{bmatrix} \times \begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}
\]

Multiplying these matrices:
- First row, first column: \( (3 \times 2) + (-5 \times 1) = 6 - 5 = 1 \)
- First row, second column: \( (3 \times 5) + (-5 \times 3) = 15 - 15 = 0 \)
- Second row, first column: \( (-1 \times 2) + (2 \times 1) = -2 + 2 = 0 \)
- Second row, second column: \( (-1 \times 5) + (2 \times 3) = -5 + 6 = 1 \)

Thus,

\[
BA = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

Again, we obtain the identity matrix \( I \).

### Conclusion:
Since both \( AB = I \) and \( BA = I \), we have shown that \( AB = BA = I \).

---

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. What is the significance of the identity matrix in linear algebra?
2. Can you always multiply two matrices in both orders, or is the product usually non-commutative?
3. How do we determine if a matrix is invertible?
4. What properties must two matrices satisfy to commute under multiplication (i.e., \( AB = BA \))?
5. How can the determinant of a matrix be used to verify if a matrix is invertible?

**Tip:** When dealing with matrix multiplication, ensure that the number of columns in the first matrix matches the number of rows in the second matrix to make the operation valid.

If A = [[2, 5], [1, 3]] and B = [[3, -5], [-1, 2]], show that AB = BA = I.

Solve the matrix problem where A = [[2, 5], [1, 3]] and B = [[3, -5], [-1, 2]], showing that AB = BA = I. Learn the steps of matrix multiplication and understand why the result is the identity matrix. Suitable for high school students in Grades 10-12.

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