" Matrix "[[a,b,(a alpha-b)],[b,c,(b alpha-c)],[2,1,0]]" Is random यदि- "

Matrix [(a,b,(a alpha-b)),(b,c,(balpha-c)),(2,1,0)] is non invertible if

If det[[a,b,a alpha+bb,c,b alpha+ca alpha+b,b alpha+c,0]]=0 then

If the determinant |[a,b,2a alpha+3b],[b,c,2b alpha+3 c],[2a alpha+3b,2b alpha+3c,0]|=0 then

If |(a,b,a alpha+b),(b,c, b alpha+c),(a alpha+b, b alpha+c,0)|=0 then

If |{:(a,b,a alpha+b),(b,c,b alpha+c),(a alpha +b,b alpha+c,0):}|=0 Prove that a,b,c are in G.P. or alpha is a root of ax^2 + 2bx + c=0

Show that the matrix A = [[1 , a,alpha , aalpha],[1, b, beta, b beta ],[1 ,c,gamma ,cgamma ]] is of renk 3 provided no two of a, b, c are equal and no two of alpha ,beta,gamma are equal.

Show that the matrix A = [[1 , a,alpha , aalpha],[1, b, beta, b beta ],[1 ,c,gamma ,cgamma ]] is of renk 3 provided no two of a, b, c are equal and no two of alpha ,beta,gamma are equal.

Show that the matrix A = [[1 , a,alpha , aalpha],[1, b, beta, b beta ],[1 ,c,gamma ,cgamma ]] is of renk 3 provided no two of a, b, c are equal and no two of alpha ,beta,gamma are equal.

Given that a alpha^(2)+2b alpha+c!=0 and that the system of equations (a alpha+b)x+alpha y+bz=0,(b alpha+c)x+by+cz=0,(a alpha+b)y+(b alpha+c)z=0 has a non trivial solution,then a ,b,c are in