Deriving the Nature of AB - BA for Symmetric Matrices A and B

This explanation explores the properties of matrices resulting from operations involving symmetric matrices. Specifically, it determines the type of matrix formed by the expression AB - BA when both A and B are symmetric matrices of the same size.

Understanding Symmetric Matrices

A fundamental concept here is the definition of a symmetric matrix. A square matrix is called symmetric if it remains unchanged when its rows and columns are interchanged. In mathematical terms, a matrix '$A$' is symmetric if its transpose, denoted as $A^T$, is equal to the matrix itself:

$$ A^T = A $$

The problem states that '$A$' and '$B$' are symmetric matrices of the same order. This implies two key conditions:

• $A^T = A$
• $B^T = B$

Key Properties of Matrix Transposition

To solve this problem, we utilize standard properties related to the transpose of matrices:

• Transpose of a Difference: The transpose of the difference between two matrices is the difference of their transposes. If '$X$' and '$Y$' are matrices, then (X - Y)^T = X^T - Y^T.
• Transpose of a Product: The transpose of the product of two matrices is the product of their transposes, but in the reverse order. If '$X$' and '$Y$' are matrices, then (XY)^T = Y^T X^T.

Step-by-Step Analysis of AB - BA

Let's consider the matrix $C = AB - BA$. Our goal is to determine whether $C$ is symmetric, skew-symmetric, or another type of matrix.

Step 1: Find the transpose of C.

We start by calculating $C^T$:

$$ C^T = (AB - BA)^T $$

Step 2: Apply the transpose property for differences.

Using the rule (X - Y)^T = X^T - Y^T, we get:

$$ C^T = (AB)^T - (BA)^T $$

Step 3: Apply the transpose property for products.

Using the rule (XY)^T = Y^T X^T, we transform each term:

$$ C^T = (B^T A^T) - (A^T B^T) $$

Step 4: Utilize the symmetric property of A and B.

Since '$A$' and '$B$' are given as symmetric matrices, we substitute $A^T = A$ and $B^T = B$ into the equation:

$$ C^T = (B A) - (A B) $$

$$ C^T = BA - AB $$

Step 5: Compare $C^T$ with $C$.

We now compare the expression for $C^T$ with the original expression for $C$:

Original matrix: $C = AB - BA$

Transposed matrix: $C^T = BA - AB$

Notice that $BA - AB$ is the negative of $AB - BA$. Therefore, we can write:

$$ C^T = -(AB - BA) $$

Which simplifies to:

$$ C^T = -C $$

Identifying the Resulting Matrix Type

A matrix '$C$' is classified as skew-symmetric if its transpose equals its negative, meaning $C^T = -C$.

Our calculation clearly shows that $(AB - BA)^T = -(AB - BA)$. This condition perfectly matches the definition of a skew-symmetric matrix.

Final Conclusion

When '$A$' and '$B$' are symmetric matrices of the same order, the matrix resulting from the operation AB - BA is always a skew-symmetric matrix.