What kind of matrix is` (AB'-BA')`. If A and B are symmetric matrices of the same order ?

Options

A. symmetric matrix
B. Skew-symmetric matrix
C. Diagonal matrix
D. Identity matrix

Answer

To determine what kind of matrix \( AB' - BA' \) is, given that \( A \) and \( B \) are symmetric matrices of the same order, we can follow these steps:

### Step 1: Understand the properties of symmetric matrices
A matrix \( A \) is symmetric if \( A' = A \). Similarly, for matrix \( B \), we have \( B' = B \).

**Hint:** Recall that a symmetric matrix is equal to its transpose.

### Step 2: Write the expression and apply the transpose
We need to analyze the expression \( AB' - BA' \). Since \( A \) and \( B \) are symmetric, we can rewrite the expression as:
\[ AB' - BA' = AB - BA \]

**Hint:** Substitute \( B' \) and \( A' \) with \( B \) and \( A \) respectively since they are symmetric.

### Step 3: Take the transpose of the expression
Now, we will take the transpose of the entire expression:
\[ (AB - BA)' = (AB)' - (BA)' \]

Using the property of transposes, we have:
\[ (AB)' = B'A' = BA \quad \text{and} \quad (BA)' = A'B' = AB \]

**Hint:** Remember the property that \( (XY)' = Y'X' \) for any matrices \( X \) and \( Y \).

### Step 4: Substitute back into the expression
Substituting these results back, we get:
\[ (AB - BA)' = BA - AB \]

### Step 5: Analyze the result
Now, we can see that:
\[ (AB - BA)' = -(AB - BA) \]
This means that \( AB - BA \) is equal to the negative of its transpose.

### Conclusion
Since \( AB - BA \) is equal to the negative of its transpose, it is a **skew-symmetric matrix**.

**Final Answer:** The matrix \( AB' - BA' \) is a skew-symmetric matrix.

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