`A` is a matrix of `3xx3` and `a_(ij)` is its elements of `i^(th)` row and `j^(th)` column. If `a_(ij)+a_(jk)+a_(ki)=0` holds for all `1 le i, j, kle 3` then

Options

A. `A` is a non singular matrix
B. `A` is a singular matrix
C. `sum_(1 let i, j le 3)a_(ij)` is equal to zero
D. `A` is a symmetric matrix

Explanation

To solve the problem, we need to analyze the condition given for the elements of the matrix \( A \) of order \( 3 \times 3 \). The condition states that:

\[ a_{ij} + a_{jk} + a_{ki} = 0 \]

holds for all \( 1 \leq i, j, k \leq 3 \).

### Step-by-Step Solution:

1. **Substituting Values**:
Let's start by substituting specific values for \( i, j, k \).

- For \( i = 1, j = 1, k = 1 \):
\[ a_{11} + a_{11} + a_{11} = 0 \implies 3a_{11} = 0 \implies a_{11} = 0 \]

- For \( i = 2, j = 2, k = 2 \):
\[ a_{22} + a_{22} + a_{22} = 0 \implies 3a_{22} = 0 \implies a_{22} = 0 \]

- For \( i = 3, j = 3, k = 3 \):
\[ a_{33} + a_{33} + a_{33} = 0 \implies 3a_{33} = 0 \implies a_{33} = 0 \]

Thus, we have:
\[ a_{11} = 0, \quad a_{22} = 0, \quad a_{33} = 0 \]

2. **Using Other Combinations**:
Now, let’s use other combinations of \( i, j, k \):

- For \( i = 1, j = 2, k = 3 \):
\[ a_{12} + a_{23} + a_{31} = 0 \implies a_{12} + a_{23} = -a_{31} \]

- For \( i = 1, j = 3, k = 2 \):
\[ a_{13} + a_{32} + a_{21} = 0 \implies a_{13} + a_{32} = -a_{21} \]

- For \( i = 2, j = 1, k = 3 \):
\[ a_{21} + a_{13} + a_{32} = 0 \implies a_{21} + a_{13} = -a_{32} \]

- For \( i = 2, j = 3, k = 1 \):
\[ a_{23} + a_{31} + a_{12} = 0 \implies a_{23} + a_{31} = -a_{12} \]

- For \( i = 3, j = 1, k = 2 \):
\[ a_{31} + a_{12} + a_{23} = 0 \implies a_{31} + a_{12} = -a_{23} \]

- For \( i = 3, j = 2, k = 1 \):
\[ a_{32} + a_{21} + a_{13} = 0 \implies a_{32} + a_{21} = -a_{13} \]

3. **Setting Up the Matrix**:
From the above equations, we can express the elements of the matrix in terms of each other. For example:
- From \( a_{12} + a_{23} = -a_{31} \), we can denote \( a_{12} = x \), \( a_{23} = y \), and \( a_{31} = -x - y \).
- Continuing this process will yield relationships among all the elements.

4. **Conclusion**:
After analyzing all combinations, we find that the only solution that satisfies all equations is when all elements of the matrix are zero. Therefore, the matrix \( A \) is a singular matrix.

### Final Result:
The matrix \( A \) is given by:

\[ A = \begin{pmatrix} 0 & a_{12} & a_{13} \\ a_{21} & 0 & a_{23} \\ a_{31} & a_{32} & 0 \end{pmatrix} \]

Where \( a_{ij} \) must satisfy the relationships derived above, leading to the conclusion that the determinant of \( A \) is zero, confirming it is a singular matrix.