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Let A = [a_ij] be an n x n matrix where a_ij = (i² + j² - i*j)(j - i), and n is odd. Which of the following can be the value of Tr(A) (the trace of A)?

1. 0
2. |A|
3. 3|A|
4. 2|A| + 3

Correct answer: 0

Solution

The diagonal entries of A are a_ii = (i² + i² - i*i)(i - i) = (i²)(0) = 0, so Tr(A) = 0. Also, a_ij + a_ji = (i²+j²-ij)(j-i) + (i²+j²-ij)(i-j) = 0, confirming A is skew-symmetric. Tr(A) = 0.

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