Matrices & Determinants
Trace of a Matrix
GRB_1000_MCQ
Grade Class 12
Question:
If $A = [a_{ij}]_{n \times n}$ and $a_{ij} = (i^2 + j^2 - ij)(j - i)$, where $n$ is odd, then the value of $tr.(A)$ is equal to:
Step-by-Step Solution
Step 1: Evaluate the diagonal entries of $A$. For diagonal entries, $i = j$, so:
$$a_{ii} = (i^2 + i^2 - i \cdot i)(i - i) = (i^2)(0) = 0$$
Step 2: Conclude the trace. Since every diagonal entry $a_{ii} = 0$:
$$tr.(A) = \sum_{i=1}^{n} a_{ii} = 0$$
Correct Answer: 1