Question

If $A$ and $B$ are symmetric matrices of the same order, then $AB - BA$ is:

• A. Symmetric matrix
• B. Zero matrix
• C. Skew-symmetric matrix
• D. Identity matrix

Hint:

When working with matrix expressions involving products, it’s essential to understand how transposition interacts with multiplication. In the case of symmetric matrices, the key property \(A^T = A\) and \(B^T = B\) helps simplify complex matrix expressions. By taking the transpose of \(AB - BA\), we can identify whether the result is symmetric or skew-symmetric based on the behavior of its transpose. A result of \( (AB - BA)^T = -(AB - BA) \) directly indicates that the matrix is skew-symmetric.

Answer 1

The Correct Option is C

To determine the nature of \(AB - BA\), let’s use the properties of symmetric and skew-symmetric matrices.

Symmetric Matrix Property: A matrix \(M\) is symmetric if \(M^T = M\).

Since \(A\) and \(B\) are symmetric matrices, we know \(A^T = A\) and \(B^T = B\).

Now, consider \((AB - BA)^T\):

\[ (AB - BA)^T = B^T A^T - A^T B^T \]

Since \(A^T = A\) and \(B^T = B\), this becomes:

\[ (AB - BA)^T = BA - AB = -(AB - BA) \]

This result implies that \(AB - BA\) is a skew-symmetric matrix, as \((AB - BA)^T = -(AB - BA)\).

Thus, \(AB - BA\) is skew-symmetric.

Answer 2

To determine the nature of \(AB - BA\), let’s explore the properties of symmetric and skew-symmetric matrices in detail. 

Symmetric Matrix Property: A matrix \(M\) is said to be symmetric if it satisfies the condition \(M^T = M\), where \(M^T\) is the transpose of matrix \(M\). In other words, a symmetric matrix remains unchanged when its rows are swapped with its columns.

Given that both matrices \(A\) and \(B\) are symmetric, we know from the definition that:

\[ A^T = A \quad \text{and} \quad B^T = B \]

This property is crucial for understanding how the matrix expression \(AB - BA\) behaves when we compute its transpose.

Step 1: Take the transpose of the expression \(AB - BA\):

Consider the transpose of \(AB - BA\), which is denoted by \((AB - BA)^T\). Using the property of transposition, we get:

\[ (AB - BA)^T = B^T A^T - A^T B^T \]

Step 2: Apply the symmetry properties of \(A\) and \(B\):

Since both \(A\) and \(B\) are symmetric matrices, we know that \(A^T = A\) and \(B^T = B\). Substituting these into the equation, we get:

\[ (AB - BA)^T = BA - AB \]

This simplifies to:

\[ (AB - BA)^T = -(AB - BA) \]

Step 3: Interpret the result:

The result \((AB - BA)^T = -(AB - BA)\) implies that \(AB - BA\) is a skew-symmetric matrix. By definition, a matrix is skew-symmetric if its transpose equals the negative of the matrix itself. In other words, a matrix \(M\) is skew-symmetric if \(M^T = -M\), which is exactly the case here.

Conclusion: Therefore, based on the transposition and the properties of symmetric matrices, we conclude that \(AB - BA\) is a skew-symmetric matrix.