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

     \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?

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.