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We are asked to find all matrices \( B = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) that commute with the matrix \( A = \begin{bmatrix} -320 & -192 \\ 216 & 208 \end{bmatrix} \). This means we need to solve for the values of \( a, b, c, d \) such that \( AB = BA \).

### Step-by-Step Explanation:

1. **Compute \( AB \):**
   \[
   AB = \begin{bmatrix} -320 & -192 \\ 216 & 208 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix}
   = \begin{bmatrix} -320a - 192c & -320b - 192d \\ 216a + 208c & 216b + 208d \end{bmatrix}
   \]

2. **Compute \( BA \):**
   \[
   BA = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} -320 & -192 \\ 216 & 208 \end{bmatrix}
   = \begin{bmatrix} a(-320) + b(216) & a(-192) + b(208) \\ c(-320) + d(216) & c(-192) + d(208) \end{bmatrix}
   = \begin{bmatrix} -320a + 216b & -192a + 208b \\ -320c + 216d & -192c + 208d \end{bmatrix}
   \]

3. **Set the two results equal to each other (since \( AB = BA \)):**
   \[
   \begin{bmatrix} -320a - 192c & -320b - 192d \\ 216a + 208c & 216b + 208d \end{bmatrix}
   =
   \begin{bmatrix} -320a + 216b & -192a + 208b \\ -320c + 216d & -192c + 208d \end{bmatrix}
   \]

4. **Form a system of equations by comparing corresponding elements:**
   - For the top-left entry: \( -320a - 192c = -320a + 216b \) 
     \[
     \Rightarrow -192c = 216b \quad \text{or} \quad c = -\frac{9}{8}b
     \]
   
   - For the top-right entry: \( -320b - 192d = -192a + 208b \)
     \[
     \Rightarrow -320b - 192d = -192a + 208b
     \quad \text{or} \quad -192d = -192a + 528b
     \quad \Rightarrow d = a - \frac{11}{4}b
     \]
   
   - For the bottom-left entry: \( 216a + 208c = -320c + 216d \)
     Using \( c = -\frac{9}{8}b \) and \( d = a - \frac{11}{4}b \), this equation simplifies, but confirms consistency in terms of \( a \) and \( b \).

   - For the bottom-right entry: \( 216b + 208d = -192c + 208d \) simplifies to a tautology, meaning no new information is added.

### General Solution:
The matrix \( B \) has two free parameters, \( a \) and \( b \), and the other elements can be expressed in terms of them:
- \( c = -\frac{9}{8}b \)
- \( d = a - \frac{11}{4}b \)

### Vector Form:
The answer in vector form is:
\[
\begin{bmatrix} a \\ b \\ -\frac{9}{8}b \\ a - \frac{11}{4}b \end{bmatrix} = s \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} + t \begin{bmatrix} 0 \\ 1 \\ -\frac{9}{8} \\ -\frac{11}{4} \end{bmatrix}
\]
where \( s \) and \( t \) are free parameters.

Would you like further clarification or more details on any step?

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Here are five related questions you might find useful:
1. How can I find the inverse of a matrix?
2. What are the conditions for two matrices to commute?
3. How does one solve a system of linear equations derived from matrices?
4. What is the significance of matrix commutativity in linear algebra?
5. How can eigenvalues help in finding commuting matrices?

**Tip:** When working with matrix equations, always check for consistency across all rows and columns to ensure the system is fully solved.

Find all the matrices B = [a b; c d] that commute with the matrix A = [-320 -192; 216 208]. Give your answer in vector form.

Learn how to find matrices that commute with a given matrix A = [-320 -192; 216 208]. This problem covers key concepts in matrix algebra and commutativity, leading to a vector-form solution.

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