Find 2×2 matrices a,b,c for which ca=bc where a≠b

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.

Explore how to find 2×2 matrices A, B, and C such that CA = BC with A ≠ B. Includes detailed steps and example matrices to satisfy the condition CA = BC. Suitable for undergraduate level study of matrix algebra and commutative properties.

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