If A and B are symmetric matrices, prove that AB BA is a skew symmetric matrix.

Explanation

To prove that \( AB - BA \) is a skew-symmetric matrix given that \( A \) and \( B \) are symmetric matrices, we will follow these steps:

### Step 1: Recall the definition of symmetric matrices
A matrix \( A \) is symmetric if \( A^T = A \) and a matrix \( B \) is symmetric if \( B^T = B \).

### Step 2: Write down the transpose of the expression \( AB - BA \)
We need to find the transpose of the expression \( AB - BA \):
\[ (AB - BA)^T = (AB)^T - (BA)^T \]

### Step 3: Apply the property of transposes
Using the property that \( (XY)^T = Y^T X^T \) for any two matrices \( X \) and \( Y \), we can rewrite the transposes:
\[ (AB)^T = B^T A^T \quad \text{and} \quad (BA)^T = A^T B^T \]

### Step 4: Substitute the transposes into the expression
Substituting the results from Step 3 into our expression from Step 2, we get:
\[ (AB - BA)^T = B^T A^T - A^T B^T \]

### Step 5: Use the fact that \( A \) and \( B \) are symmetric
Since \( A \) and \( B \) are symmetric matrices, we have:
\[ B^T = B \quad \text{and} \quad A^T = A \]
Substituting these into our expression gives:
\[ (AB - BA)^T = B A - A B \]

### Step 6: Rearranging the expression
Notice that:
\[ (AB - BA)^T = BA - AB = -(AB - BA) \]

### Step 7: Conclusion
Since we have shown that:
\[ (AB - BA)^T = -(AB - BA) \]
This means that \( AB - BA \) is skew-symmetric.

### Final Statement
Thus, we conclude that \( AB - BA \) is a skew-symmetric matrix.

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