Lat`A = [a_(ij)]_(3xx 3).` If tr is arithmetic mean of elements of rth row and `a_(ij )+ a_( jk) + a_(ki)=0` holde for all `1 le i, j, k le 3.` ` sum_(1lei) sum_(jle3) a _(ij)` is not equal to

Options

A. `t_(1) + t_(2) + t_(3)`
B. zero
C. `(det(A))^(2)`
D. `t_(1) t_(2)t_(3)`

Explanation

To solve the problem step by step, we need to analyze the given conditions and compute the required summation.

### Step 1: Understand the Matrix and Given Conditions
We have a 3x3 matrix \( A = [a_{ij}] \). The conditions provided are:
1. The trace \( \text{tr} \) is the arithmetic mean of the elements of the r-th row.
2. The equation \( a_{ij} + a_{jk} + a_{ki} = 0 \) holds for all \( 1 \leq i, j, k \leq 3 \).

### Step 2: Calculate the Summation
We need to evaluate the summation \( \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} \).

This can be expanded as:
\[ \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} = a_{11} + a_{12} + a_{13} + a_{21} + a_{22} + a_{23} + a_{31} + a_{32} + a_{33} \]

### Step 3: Grouping the Terms
We can group the terms based on the rows:
- Row 1: \( a_{11} + a_{12} + a_{13} \)
- Row 2: \( a_{21} + a_{22} + a_{23} \)
- Row 3: \( a_{31} + a_{32} + a_{33} \)

Thus, we can write:
\[ \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} = (a_{11} + a_{12} + a_{13}) + (a_{21} + a_{22} + a_{23}) + (a_{31} + a_{32} + a_{33}) \]

### Step 4: Applying the Condition \( a_{ij} + a_{jk} + a_{ki} = 0 \)
From the condition \( a_{ij} + a_{jk} + a_{ki} = 0 \), we can derive that:
- For \( i=1, j=1, k=2 \): \( a_{11} + a_{12} + a_{21} = 0 \)
- For \( i=1, j=2, k=3 \): \( a_{12} + a_{23} + a_{32} = 0 \)
- For \( i=2, j=1, k=3 \): \( a_{21} + a_{31} + a_{13} = 0 \)

This implies that the sum of the elements in each row is related to the others.

### Step 5: Finding the Trace
The trace of the matrix is given by:
\[ \text{tr}(A) = a_{11} + a_{22} + a_{33} \]
Since the trace is the arithmetic mean of the elements of each row, we can express:
\[ \text{tr}(A) = \frac{1}{3} \left( (a_{11} + a_{12} + a_{13}) + (a_{21} + a_{22} + a_{23}) + (a_{31} + a_{32} + a_{33}) \right) \]

### Step 6: Conclusion
Since we have established that the sum of the elements in the matrix is influenced by the conditions given, we can conclude that the sum \( \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} \) is likely to equal zero based on the relationships established.

### Final Answer
Thus, the summation \( \sum_{i=1}^{3} \sum_{j=1}^{3} a_{ij} \) is not equal to a specific value, and we conclude that it is equal to zero.