If A is a skew symmetric matrix of order `n` and C is a column matrix of order `nxx1`, then `C^(T)AC` is

Options

A. an identity matrix of ordern
B. an identity of order 1
C. a zero matrix of order 1
D. None of the above

Explanation

To solve the problem, we need to analyze the expression \( C^T A C \) where \( A \) is a skew-symmetric matrix of order \( n \) and \( C \) is a column matrix of order \( n \times 1 \).

### Step-by-Step Solution:

1. **Understanding Skew-Symmetric Matrix**:
A matrix \( A \) is skew-symmetric if \( A^T = -A \). This means that the transpose of the matrix is equal to the negative of the matrix itself.

2. **Transpose of the Expression**:
We start with the expression \( C^T A C \). We will take the transpose of this expression:
\[ (C^T A C)^T \]
Using the property of transposes, we know that \( (XYZ)^T = Z^T Y^T X^T \). Thus, we can write:
\[ (C^T A C)^T = C^T A^T (C^T)^T = C^T A^T C \]
Since \( (C^T)^T = C \), we can simplify this to:
\[ (C^T A C)^T = C^T A^T C \]

3. **Substituting the Skew-Symmetric Property**:
Now, substituting \( A^T = -A \) into our expression:
\[ (C^T A C)^T = C^T (-A) C = -C^T A C \]

4. **Setting Up the Equation**:
Let \( K = C^T A C \). From the previous step, we have:
\[ K^T = -K \]
This indicates that \( K \) is equal to its own negative.

5. **Conclusion**:
The only matrix that is equal to its own negative is the zero matrix. Therefore, we conclude:
\[ K = 0 \]
Hence, we have:
\[ C^T A C = 0 \]

### Final Answer:
Thus, \( C^T A C \) is a zero matrix of order \( 1 \).