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The problem asks for the values of \( k \), if any, that will make the matrix multiplication \( AB = BA \), where

\[
A = \begin{bmatrix} 2 & 5 \\ -3 & 1 \end{bmatrix}
\quad \text{and} \quad
B = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}.
\]

### Step 1: Calculate \( AB \)
\[
AB = \begin{bmatrix} 2 & 5 \\ -3 & 1 \end{bmatrix} \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix}
= \begin{bmatrix} 2(4) + 5(3) & 2(-5) + 5(k) \\ -3(4) + 1(3) & -3(-5) + 1(k) \end{bmatrix}
= \begin{bmatrix} 8 + 15 & -10 + 5k \\ -12 + 3 & 15 + k \end{bmatrix}
= \begin{bmatrix} 23 & -10 + 5k \\ -9 & 15 + k \end{bmatrix}.
\]

### Step 2: Calculate \( BA \)
\[
BA = \begin{bmatrix} 4 & -5 \\ 3 & k \end{bmatrix} \begin{bmatrix} 2 & 5 \\ -3 & 1 \end{bmatrix}
= \begin{bmatrix} 4(2) + (-5)(-3) & 4(5) + (-5)(1) \\ 3(2) + k(-3) & 3(5) + k(1) \end{bmatrix}
= \begin{bmatrix} 8 + 15 & 20 - 5 \\ 6 - 3k & 15 + k \end{bmatrix}
= \begin{bmatrix} 23 & 15 \\ 6 - 3k & 15 + k \end{bmatrix}.
\]

### Step 3: Set \( AB = BA \)
For \( AB = BA \), the corresponding elements of the two matrices must be equal:

1. \( 23 = 23 \) (This is always true, so no information about \( k \) from here.)
2. \( -10 + 5k = 15 \)
3. \( -9 = 6 - 3k \)
4. \( 15 + k = 15 + k \) (This is also always true, so no information about \( k \) here either.)

### Step 4: Solve the equations
From equation 2:
\[
-10 + 5k = 15 \quad \Rightarrow \quad 5k = 25 \quad \Rightarrow \quad k = 5.
\]

From equation 3:
\[
-9 = 6 - 3k \quad \Rightarrow \quad -15 = -3k \quad \Rightarrow \quad k = 5.
\]

Thus, \( k = 5 \) is the only value that satisfies both equations.

### Conclusion:
The value of \( k \) that makes \( AB = BA \) is \( k = 5 \).

---

Do you have any questions or would you like more details on any step? Here are five related questions to expand on this concept:

1. How do you determine if two general square matrices commute?
2. Can there be more than one value of \( k \) that satisfies \( AB = BA \)?
3. What is the significance of matrix multiplication in solving systems of linear equations?
4. How can matrix properties like commutativity affect transformations in computer graphics?
5. What are some real-world applications of matrix commutation?

**Tip:** When multiplying matrices, ensure that the inner dimensions match (e.g., an \( m \times n \) matrix with an \( n \times p \) matrix) to obtain a valid product.

Let A = [[2, 5], [-3, 1]] and B = [[4, -5], [3, k]]. What value(s) of k, if any, will make AB = BA?

Learn how to determine the value of k that makes the matrices A and B commute, where A = [[2, 5], [-3, 1]] and B = [[4, -5], [3, k]]. Step-by-step solution involving matrix multiplication and comparison of elements.

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