Understanding Symmetric and Skew-Symmetric Matrices

This solution explores the properties of matrices derived from combining a symmetric matrix and a skew-symmetric matrix of the same order. We focus on determining the nature of the matrix $AB - BA$.

First, let's clarify the definitions:

• A matrix $M$ is defined as symmetric if its transpose ($M^T$) is identical to the matrix itself. The condition is $M^T = M$.
• A matrix $M$ is defined as skew-symmetric if its transpose ($M^T$) is the negative of the matrix. The condition is $M^T = -M$.

In this problem, we are given:

• Matrix $A$ is symmetric, so $A^T = A$.
• Matrix $B$ is skew-symmetric, so $B^T = -B$.
• Matrices $A$ and $B$ have the same order (dimensions).

Analyzing the Matrix Expression $AB - BA$

Our goal is to determine the characteristic property (symmetric, skew-symmetric, etc.) of the matrix formed by the expression $AB - BA$. Let's denote this resulting matrix as $C$:

$C = AB - BA$

To classify matrix $C$, we will compute its transpose, $C^T$, and compare it with $C$ and $-C$.

Step-by-Step Calculation of the Transpose $C^T$

We use fundamental rules of matrix transposition:

• The transpose of a sum or difference of matrices is the sum or difference of their transposes: $(X \pm Y)^T = X^T \pm Y^T$.
• The transpose of a product of matrices is the product of their transposes in reverse order: $(XY)^T = Y^T X^T$.

Let's find the transpose of $C$:

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

Applying the rule for the transpose of a difference:

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

Now, applying the rule for the transpose of a product to each term:

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

Applying the Given Conditions to $C^T$

Substitute the properties of matrices $A$ and $B$ (i.e., $A^T = A$ and $B^T = -B$) into the expression for $C^T$:

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

Perform the matrix multiplications:

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

Simplify the expression:

$C^T = -BA + AB$

Rearrange the terms to match the original form of $C$:

$C^T = AB - BA$

Determining the Nature of $AB - BA$

Our calculation shows that $C^T = AB - BA$. Since we initially defined $C = AB - BA$, we have found that:

$C^T = C$

By definition, a matrix whose transpose is equal to itself is a symmetric matrix.

Therefore, the expression $AB - BA$ results in a symmetric matrix.

Evaluating the Options

Based on our derivation, let's consider the given options:

• Skew-symmetric: This holds if $C^T = -C$. Our finding is $C^T = C$, so this is generally incorrect.
• Symmetric: This holds if $C^T = C$. Our finding directly matches this definition.
• Diagonal: A diagonal matrix is a specific type of symmetric matrix where off-diagonal elements are zero. While $AB - BA$ might be diagonal in certain cases, it is always symmetric, which is a broader and guaranteed property.
• Zero matrix: This is the matrix where all elements are zero. The zero matrix is both symmetric and skew-symmetric. $AB - BA$ could be the zero matrix for specific $A$ and $B$, but it is not true for all possible symmetric $A$ and skew-symmetric $B$. The general property is symmetry.

The analysis confirms that $AB - BA$ is always a symmetric matrix, given that $A$ is symmetric and $B$ is skew-symmetric of the same order.