Determinants Matrices Question 68
Elements of a matrix A of order $ 10\times 10 $ are defined as $ a _{ij}={w^{i+j}} $ (where w is cube root of unity), then trace of the matrix is
Options:
A) 0
B) 1
C) 3
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
[d] $ tr(A)=\sum\limits _{i=1}^{n}{a _{ii}}=(a _{11}+a _{22}+a _{33}+….+a _{nn}) $
$ =(w^{2}+w^{4}+w^{6}+….+w^{20})= $
$ w^{2}(1+w^{2}+w^{4}+…+w^{18}) $
$ =w^{2}[(1+w+w^{2})+…+(1+w+w^{2})+1] $
$ =w^{2}\times 1=w^{2} $
BETA
