To find $2 \times 2$ matrices $A$, $B$, and $C$ such that $CA = BC$ and $A \neq B$, we can approach this problem by constructing an example with specific matrices.

Let's take:

$A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Now let's verify if these matrices satisfy the equation $CA = BC$:

1. Compute $CA$: $CA = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$

2. Compute $BC$: $BC = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$

Since $CA = BC = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, the condition $CA = BC$ is satisfied. Furthermore, $A \neq B$, as required.

Summary

Thus, the matrices $A$, $B$, and $C$ given by:

$A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

satisfy $CA = BC$ with $A \neq B$.

Would you like further details or have any other questions?

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Here are 5 related questions to expand on this:

1. What is the significance of the condition $A \neq B$ in matrix equations?
2. Can you generalize the condition $CA = BC$ for larger matrices?
3. How does the commutative property apply to matrix multiplication?
4. What are the eigenvalues of the matrices $A$ and $B$ in this example?
5. Can this problem be solved if matrices $A$, $B$, and $C$ are diagonal?

Tip: When dealing with matrix equations, always verify the dimensions of matrices before performing operations like multiplication or addition.