How do you prove that AB=BA if and only if AB is also symmetric?

To show that \( AB = BA \) if and only if \( AB \) is symmetric, we need to explore both directions of the statement.

Understanding Symmetric Matrices

A matrix \( M \) is symmetric if \( M^T = M \), where \( M^T \) denotes the transpose of \( M \). For our case, we will analyze the product of two matrices \( A \) and \( B \).

Proving the Forward Direction

Assume \( AB \) is symmetric. This means:

• We have \( (AB)^T = AB \).
• Using the property of transposes, we know \( (AB)^T = B^T A^T \).

Thus, we can write:

B^T A^T = AB

Now, if we multiply both sides by \( A^{-1} \) (assuming \( A \) is invertible), we get:

B^T = A B A^{-1}

For this to hold for all \( A \) and \( B \), it implies that \( A \) and \( B \) must commute, leading us to \( AB = BA \).

Proving the Reverse Direction

Now, let’s assume \( AB = BA \). We want to show that \( AB \) is symmetric:

• Since \( AB = BA \), we can take the transpose:
• Thus, \( (AB)^T = (BA)^T = A^T B^T \).

Given that \( A \) and \( B \) commute, we have:

A^T B^T = AB

This shows that \( AB \) is symmetric because \( (AB)^T = AB \).

Final Thoughts

In summary, we have demonstrated that \( AB = BA \) if and only if \( AB \) is symmetric through logical deductions and properties of matrix operations. This relationship highlights the importance of symmetry in matrix algebra.