For a real number `alpha,` if the system `[[1,alpha,alpha^2],[alpha,1,alpha],[alpha^2,alpha,1]][[x],[ y], [z]]=[[1],[-1],[1]]` of linear equations, has infinitely many solutions, then `1+alpha+alpha^2=`

If f(alpha)=[[1,alpha,alpha^2],[alpha,alpha^2,1],[alpha^2,1,alpha]] then find the value of f(3^(1/3))

If alpha is a non-real cube root of -2 , then the value of |(1,2 alpha,1),(alpha^(2),1,3 alpha^(2)),(2,2 alpha,1)| , is

If alpha is a non real root of x^(6)=1 then alpha^(5)+alpha^(3)+alpha+1hline alpha^(2)+1

Let A(alpha)=[(cos alpha, 0,sin alpha),(0,1,0),(sin alpha, 0, cos alpha)] and [(x,y,z)]=[(0,1,0)] . If the system of equations has infinite solutions and sum of all the possible value of alpha in [0, 2pi] is kpi , then the value of k is equal to

For real numbers alpha and beta , consider the following system of linear equations: x+y-z=2, x+2y+alpha z=1, 2x- y+z= beta . If the system has infinite solutions, then alpha+beta is equal to ____

If alpha is a non-real root of x^(6)=1 then (alpha^(5)+alpha^(3)+alpha+1)/(alpha^(2)+1)=

The system of equations alpha x-y-z=alpha-1 , x-alpha y-z=alpha-1,x-y-alpha z=alpha-1 has no solution if alpha

The system of equations alphax+y+z=alpha-1, x+alphay+z=alpha-1 x+y+alphaz=alpha-1 and has no solution if alpha is