If `A` is a skew symmetric matrix of order `3`, `B` is a `3xx1` column matrix and `C=B^(T)AB`, then which of the following is false?

Options

A. C is singular
B. C is non singular
C. C is a symmetric matrix
D. C is a skew symmetric matrix

Answer

To solve the problem step by step, we need to analyze the properties of the matrices involved and determine which statement about the matrix \( C = B^T A B \) is false.

### Step 1: Understand the properties of skew-symmetric matrices
A skew-symmetric matrix \( A \) of order 3 satisfies the property:
\[ A^T = -A \]
This means that the transpose of the matrix is equal to the negative of the matrix itself.

### Step 2: Determine the dimensions of the matrices
- \( A \) is a \( 3 \times 3 \) skew-symmetric matrix.
- \( B \) is a \( 3 \times 1 \) column matrix.
- The transpose of \( B \), \( B^T \), will be a \( 1 \times 3 \) row matrix.

### Step 3: Calculate the dimensions of \( C \)
The expression for \( C \) is given as:
\[ C = B^T A B \]
- The product \( AB \) will be a \( 3 \times 1 \) matrix (since \( A \) is \( 3 \times 3 \) and \( B \) is \( 3 \times 1 \)).
- Then, \( B^T A B \) becomes \( (1 \times 3)(3 \times 1) \), resulting in a \( 1 \times 1 \) matrix.

### Step 4: Analyze the properties of \( C \)
Since \( C \) is a \( 1 \times 1 \) matrix, it can be treated as a scalar. We need to check if \( C \) is symmetric, skew-symmetric, or singular.

1. **Symmetric**: A matrix \( C \) is symmetric if \( C = C^T \). Since \( C \) is a \( 1 \times 1 \) matrix, it is trivially symmetric.

2. **Skew-symmetric**: A matrix \( C \) is skew-symmetric if \( C^T = -C \). For a \( 1 \times 1 \) matrix, this implies \( C = -C \), which leads to \( 2C = 0 \) or \( C = 0 \). Thus, \( C \) can be skew-symmetric if it is zero.

3. **Singular**: A matrix is singular if its determinant is zero. Since \( C \) is a \( 1 \times 1 \) matrix, its determinant is simply its value. If \( C = 0 \), then it is singular.

### Step 5: Conclusion
- \( C \) is symmetric (true).
- \( C \) can be skew-symmetric if \( C = 0 \) (true).
- \( C \) is singular if \( C = 0 \) (true).

Now, we need to identify which statement is false. The statements regarding \( C \) being symmetric, skew-symmetric, and singular are all true under the condition that \( C = 0 \).

### Final Answer
The false statement among the options provided is the one that claims \( C \) is not singular, as \( C \) is indeed singular when it equals zero.