Find 2x2 matrices A,B,C for which CA=BC and A≠B

To find \(2 \times 2\) matrices \( A \), \( B \), and \( C \) such that \( CA = BC \) and \( A \neq B \), we can approach the problem with some trial and error or by constructing matrices that satisfy the equation.

Let's start with some specific matrices:

Let:
\[
A = \begin{pmatrix}
1 & 0 \\
0 & 2
\end{pmatrix}, \quad 
B = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \quad 
C = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\]

Now, let's compute \( CA \) and \( BC \):

### Calculate \( CA \):
\[
CA = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & 2
\end{pmatrix}
=
\begin{pmatrix}
0 & 2 \\
1 & 0
\end{pmatrix}
\]

### Calculate \( BC \):
\[
BC = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]

We see that \( CA = BC \) is not satisfied, so we need to try another approach.

Let's redefine the matrices:

Let:
\[
A = \begin{pmatrix}
1 & 0 \\
0 & 2
\end{pmatrix}, \quad 
B = \begin{pmatrix}
2 & 0 \\
0 & 1
\end{pmatrix}, \quad 
C = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\]

Now, let's compute \( CA \) and \( BC \) again.

### Calculate \( CA \):
\[
CA = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & 2
\end{pmatrix}
=
\begin{pmatrix}
0 & 2 \\
1 & 0
\end{pmatrix}
\]

### Calculate \( BC \):
\[
BC = \begin{pmatrix}
2 & 0 \\
0 & 1
\end{pmatrix}
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
=
\begin{pmatrix}
0 & 2 \\
1 & 0
\end{pmatrix}
\]

Here, \( CA = BC \), but \( A \neq B \) since:
\[
A = \begin{pmatrix}
1 & 0 \\
0 & 2
\end{pmatrix}, \quad 
B = \begin{pmatrix}
2 & 0 \\
0 & 1
\end{pmatrix}
\]

Thus, the matrices \( A \), \( B \), and \( C \) satisfying \( CA = BC \) and \( A \neq B \) are:

\[
A = \begin{pmatrix}
1 & 0 \\
0 & 2
\end{pmatrix}, \quad 
B = \begin{pmatrix}
2 & 0 \\
0 & 1
\end{pmatrix}, \quad 
C = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How can we generalize the conditions for matrices \( A \), \( B \), and \( C \) such that \( CA = BC \) holds?
2. What happens if we consider \( C \) as a diagonal matrix?
3. Can we find a similar example with \( 3 \times 3 \) matrices?
4. What if \( C \) is the identity matrix? How does that change the relationship between \( A \) and \( B \)?
5. How do the eigenvalues of \( A \) and \( B \) relate under the given conditions?

**Tip:** When solving matrix equations, exploring specific examples can provide insights, but always check the generality of the solution to ensure it isn't just a special case.

Explore how to find specific 2x2 matrices A, B, and C satisfying the condition CA = BC where A ≠ B. Detailed examples and computations provided.

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