For any square matrix `A,(A-A')` is :

Options

A. Symmetric matrix
B. Skew-symmetric matrix
C. Scalar matrix
D. Zero matrix

Answer

To solve the problem of determining the nature of the matrix \( A - A' \) (where \( A' \) is the transpose of matrix \( A \)), we will follow these steps:

### Step 1: Understand the Definitions
A matrix \( A \) is called:
- **Symmetric** if \( A' = A \)
- **Skew-Symmetric** if \( A' = -A \)

### Step 2: Compute the Transpose of \( A - A' \)
We need to find the transpose of the expression \( A - A' \):
\[ (A - A')' = A' - (A')' \]
Using the property of transposes, we know that the transpose of the transpose of a matrix is the matrix itself:
\[ (A - A')' = A' - A \]

### Step 3: Rewrite the Expression
We can rewrite the expression \( A' - A \) as:
\[ A' - A = - (A - A') \]

### Step 4: Relate the Transpose to the Original Expression
From the previous step, we have:
\[ (A - A')' = - (A - A') \]
Let \( X = A - A' \). Then we can express the relationship as:
\[ X' = -X \]

### Step 5: Conclusion
The relationship \( X' = -X \) indicates that \( X \) is a skew-symmetric matrix. Therefore:
\[ A - A' \text{ is a skew-symmetric matrix.} \]

### Final Answer
The correct answer is **B: Skew-Symmetric Matrix**.

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