Home
Class 12
MATHS
If omega!=1 is a complex cube root of un...

If `omega!=1` is a complex cube root of unity, then prove that `[{:(1+2omega^(2017)+omega^(2018)," "omega^(2018),1),(1,1+2omega^(2018)+omega^(2017),omega^(2017)),(omega^(2017),omega^(2018),2+2omega^(2017)+omega^(2018)):}]`is singular

Text Solution

Verified by Experts

Let `A=[{:(1+2omega^(2017)+omega^(2018)," "omega^(2018),1),(1,1+2omega^(2018)+omega^(2017),omega^(17)),(omega^(17),omega^(18),2+2omega^(2017)+omega^(2018)):}]`
` therefore" " omega^(3)=1rArr omega^(2017)=omega`
and `omega^(2018)=omega^(2)` then
`[(1+2omega+omega^(2),omega^(2),1),(1,1+omega^(2)+2omega,omega),(omega,omega^(2),2+omega+2omega^(2))]`
`=[(omega,omega^(2),1),(1,omega,omega),(omega,omega^(2),-omega)]" " [therefore1+omega+omega^(2)=0]`
Now, `|A|= [(omega,omega^(2),1),(1,omega,omega),(omega,omega^(2),-omega)]= omega[(omega,omega,1),(1,1,omega),(omega,omega,-omega)]=0 thus, `|A|=0.` Hence, A is singular matrix.
Promotional Banner

Similar Questions

Explore conceptually related problems

If omega is a complex cube root of unity, then (1-omega+omega^(2))^(6)+(1-omega^(2)+omega)^(6)=

If omega is a complex cube root of unity then the matrix A = [(1, omega^(2),omega),(omega^(2),omega,1),(omega,1,omega^(2))] is a

omega is a complex cube root of unity,then (1-omega)(1-+-ega^(2))(1-omega^(4))(1-omega^(8))

If omega ne 1 is a cube root of unity, then 1, omega, omega^(2)

If omega is the complex cube root of unity,then prove that det[[1,1,11,-1-omega^(2),omega^(2)1,omega^(2),omega^(4)]]=+-3sqrt(3)i

If omega (ne 1) is a cube root of unity, then the value of tan[(omega^(2017) + omega^(2225)) pi - pi//3]

If omega is a complex cube root of unity, then a root of the equation |(x +1,omega,omega^(2)),(omega,x + omega^(2),1),(omega^(2),1,x + omega)| = 0 , is

If omega is a complex cube root of unity.Show that Det[[1,omega,omega^(2)omega,omega^(2),1omega^(2),1,omega]]=0