If `A=[a_(ij)]_(mxxn)` and `a_(ij)=(i^(2)+j^(2)-ij)(j-i)`, `n` odd, then which of the following is not the value of `Tr(A)`

Options

A. `0`
B. `|A|`
C. `2|A|`
D. none of these

Explanation

To solve the problem, we need to analyze the given matrix \( A = [a_{ij}]_{m \times n} \) where \( a_{ij} = (i^2 + j^2 - ij)(j - i) \) and \( n \) is odd. We will determine the trace of the matrix \( A \) and identify which of the given options is not a possible value of \( \text{Tr}(A) \).

### Step 1: Understand the matrix element \( a_{ij} \)

The element of the matrix \( A \) is defined as:
\[ a_{ij} = (i^2 + j^2 - ij)(j - i) \]

### Step 2: Check the symmetry of the matrix

To check if the matrix is skew-symmetric, we compute \( a_{ji} \):
\[ a_{ji} = (j^2 + i^2 - ji)(i - j) \]
This can be rewritten as:
\[ a_{ji} = (i^2 + j^2 - ij)(i - j) = -a_{ij} \]
This shows that \( a_{ji} = -a_{ij} \), confirming that \( A \) is a skew-symmetric matrix.

### Step 3: Properties of skew-symmetric matrices

For any skew-symmetric matrix, the trace is given by:
\[ \text{Tr}(A) = \sum_{i=1}^{m} a_{ii} \]
Since \( a_{ii} = (i^2 + i^2 - ii)(i - i) = 0 \), we find that:
\[ \text{Tr}(A) = 0 \]

### Step 4: Determine possible values of Tr(A)

Since \( A \) is skew-symmetric and \( n \) is odd, the determinant of \( A \) is zero. The determinant of a skew-symmetric matrix of odd order is always zero. Therefore, the possible values of \( \text{Tr}(A) \) are:
- Zero (since the trace of a skew-symmetric matrix is zero).

### Step 5: Identify the answer

Given the options, we need to find which of the following is not a value of \( \text{Tr}(A) \). Since we established that \( \text{Tr}(A) = 0 \), any option that is not zero is the answer.

### Conclusion

Thus, the value that is not the value of \( \text{Tr}(A) \) is any non-zero value provided in the options.