If A is a square matrix, then A is skew symmetric if

Options

A. `A^(2)=A`
B. `A^(2)=l`
C. `A^(T)=A`
D. `A^(T)=-A`

Answer

To determine the condition for a square matrix \( A \) to be skew-symmetric, we can follow these steps:

### Step 1: Definition of Skew-Symmetric Matrix
A matrix \( A \) is said to be skew-symmetric if it satisfies the condition:
\[ A^T = -A \]
where \( A^T \) is the transpose of matrix \( A \).

### Step 2: Taking the Transpose of the Matrix
Let’s denote a matrix \( C \) as:
\[ C = A - A^T \]
Now, we will take the transpose of \( C \):
\[ C^T = (A - A^T)^T \]

### Step 3: Applying Transpose Properties
Using the property of transposes, we can expand \( C^T \):
\[ C^T = A^T - (A^T)^T \]
Since the transpose of a transpose returns the original matrix, we have:
\[ C^T = A^T - A \]

### Step 4: Rearranging the Expression
We can rearrange the expression for \( C^T \):
\[ C^T = - (A - A^T) = -C \]

### Step 5: Conclusion
Since we have shown that:
\[ C^T = -C \]
This means that \( C \) is skew-symmetric. For \( A \) to be skew-symmetric, it must hold that:
\[ A^T = -A \]

### Final Answer
Thus, the condition for a square matrix \( A \) to be skew-symmetric is:
\[ A^T = -A \]