If a matrix is both symmetric matrix and skew symmetric matrix then

The correct option is "A is zero matrix".

To understand why this is correct, let us analyze the properties of symmetric and skew-symmetric matrices step-by-step.

Step 1: Definition of a Symmetric Matrix
A square matrix $A$ is symmetric if it is equal to its transpose. This is written as:

$$A^T=A$$

Step 2: Definition of a Skew-Symmetric Matrix
A square matrix $A$ is skew-symmetric if it is equal to the negative of its transpose. This is written as:

$$A^T=-A$$

Step 3: Combining the Two Conditions
Since the matrix $A$ is given to be both symmetric and skew-symmetric, we can equate the two expressions for its transpose $A^T$:

$$A=-A$$

Step 4: Solving for Matrix A
Adding matrix $A$ to both sides of the equation gives:

$$A+A=O$$

where $O$ represents the zero matrix (a matrix containing only zero elements). This simplifies to:

$$2A=O$$

Dividing by 2, we get:

$$A=O$$

Therefore, the matrix $A$ must be a zero matrix.