Thedeterminant |{:(a,b,aalpha+b),(b,c,aalpha+c),(aalpha+b,balpha+c,0):}|=0 if a,b,c are in | 12...

If |{:(a, b, aalpha+b),(b,c,balpha+c),(aalpha+b,balpha+c," "0):}|=0 then

the determinant |{:(a,,b,,aalpha+b),(b,,c,,balpha+c),(aalpha+b,,balpha+c,,0):}|=0 is equal to zero if

The determinant Delta=|{:(a,b,aalpha+c),(b,c,balpha+c),(aalpha+b,balpha+c,0):}| is equal to zero if

The determinant |a b aalpha+bb c balpha+c aalpha+bbalpha+c0|=0, if a ,b , c are in A.P. a ,b ,c are in G.P. a ,b ,c are in H.P. alpha is a root of the equation a x^2+b c+c=0 (x-alpha) is a factor of a x^2+2b x+c

If the determinant |[a,b,2aalpha+3b],[b,c,2balpha+3c],[2aalpha+3b,2balpha+3c,0]|=0 then (a)a , b , c are in H.P. alpha is root of 4a x^2+12 b x+9c=0 or a , b , c are in G.P. a , b , c , are in G.P. only a , b ,c are in A.P.

If |{:(a,b,aalpha-b),(b,c,balpha-c),(2,1," "0):}|=0andalpha!=1//2 , then a,b,c are in

The determination Delta=|{:(,b,c,balpha+c),(,c,d,calpha+d),(,balpha+c,calpha+d,aalpha^(3)-calpha):}| is equal to zero if

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