If a matrix `A` is both symmetric and skew-symmetric, then `A` is a

Options

A. Diagonal matrix
B. Zero matrix
C. Scalar matrix
D. Square matrix

Explanation

To determine the type of matrix \( A \) that is both symmetric and skew-symmetric, we can follow these steps:

### Step 1: Understand the properties of symmetric and skew-symmetric matrices.
- A matrix \( A \) is **symmetric** if \( A^T = A \).
- A matrix \( A \) is **skew-symmetric** if \( A^T = -A \).

### Step 2: Set up the equations based on the properties.
From the properties:
1. For symmetric:
\[ A^T = A \quad \text{(Equation 1)} \]
2. For skew-symmetric:
\[ A^T = -A \quad \text{(Equation 2)} \]

### Step 3: Equate the two expressions for \( A^T \).
Since both equations represent \( A^T \), we can set them equal to each other:
\[ A = -A \]

### Step 4: Solve for \( A \).
Rearranging the equation gives:
\[ A + A = 0 \implies 2A = 0 \implies A = 0 \]

### Conclusion:
Thus, if a matrix \( A \) is both symmetric and skew-symmetric, it must be the zero matrix.

### Final Answer:
The matrix \( A \) is the **zero matrix**.

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