Q. Elements of matrix $A$ of order $10 \times 10$ are defined as$a_{w} = w^{i + j}$(where $w$ is cube coot of unity), then trace ($A$) of the matrix is
A) 0
B) 1
C) 3
D) None of these
Explanation
(d)
tr$(A) = \sum_{i = j}^{}a_{ij}$
$= (a_{11} + a_{22} + a_{33} + \cdots + a_{10\infty 10})$
$= (w^{2} + w^{4} + w^{6} + \cdots + w^{20})$
$= w^{2}(1 + w^{2} + w^{4} + \cdots + w^{18})$
$= w^{2}\lbrack\left( 1 + w + w^{2} \right) + \cdots + \left( 1 + w + w^{2} \right) + 1\rbrack$
$= w^{2} \times 1$
$\Rightarrow$tr$(A) = w^{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.