If AB= A , BA=B , prove that B transpose × A transpose = A transpose 

We are given the following equations involving matrices \( A \) and \( B \):

\[
AB = A \quad \text{(1)}
\]
\[
BA = B \quad \text{(2)}
\]

We want to prove that \( B^T A^T = A^T \), where \( B^T \) and \( A^T \) are the transposes of \( B \) and \( A \), respectively.

### Step-by-Step Proof:

1. **Start by transposing both sides of equation (2):**
   \[
   BA = B
   \]
   Taking the transpose of both sides:
   \[
   (BA)^T = B^T
   \]
   Using the property of transposition that \( (XY)^T = Y^T X^T \), we have:
   \[
   A^T B^T = B^T \quad \text{(3)}
   \]

2. **Simplifying equation (3):**
   From equation (3), we now have:
   \[
   A^T B^T = B^T
   \]
   This tells us that when we multiply \( A^T \) by \( B^T \), the result is simply \( B^T \).

3. **Transpose equation (1):**
   Now, take the transpose of equation (1):
   \[
   AB = A
   \]
   Taking the transpose of both sides:
   \[
   (AB)^T = A^T
   \]
   Again using the transposition property \( (XY)^T = B^T A^T \), we get:
   \[
   B^T A^T = A^T \quad \text{(4)}
   \]

4. **Conclusion:**
   From equation (4), we see that \( B^T A^T = A^T \), which is the result we wanted to prove.

Thus, we have shown that \( B^T A^T = A^T \), as required.

### Do you have any questions or would you like further details?

Here are 5 related questions you might find interesting:
1. Can we extend the result to any two matrices \( A \) and \( B \) with similar properties?
2. How does the proof change if \( A \) or \( B \) are not square matrices?
3. What other properties can be deduced about \( A \) and \( B \) given these conditions?
4. How does the rank of \( A \) and \( B \) relate to this proof?
5. Can we derive conditions for when \( A \) and \( B \) are invertible based on these equations?

**Tip:** Transposition reverses the order of matrix multiplication, so always be mindful of the order when working with transposes!

If AB = A, BA = B, prove that B transpose × A transpose = A transpose.

Explore a detailed proof for the equation B transpose × A transpose = A transpose given AB = A and BA = B. Learn about matrix multiplication and the properties of transposition. Suitable for students in Grades 10-12.

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