Evaluate: `|[alpha, beta,gamma],[alpha^2,beta^2,gamma^2],[beta+gamma,gamma+alpha,alpha+beta]|`

Prove that, |{:(alpha,beta,gamma),(alpha^2,beta^2,gamma^2),(beta+gamma,gamma+alpha,alpha+beta):}|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

If alpha,beta,gamma are the roots of a x^3+b x^2+cx+d=0 and |[alpha,beta,gamma],[beta,gamma,alpha],[gamma,alpha,beta]|=0, alpha!=beta!=gamma then find the equation whose roots are alpha+beta-gamma,beta+gamma-alpha , and gamma+alpha-beta .

If alpha,beta,gamma are such that alpha+beta+gamma=2,""alpha^2+beta^2+gamma^2=6,""alpha^3+beta^3+gamma^3=8,t h e n""alpha^4+beta^4+gamma^4 is a. 18 b. 10 c. 15 d. 36

Using properties of determinants in Exercises prove that : {:|( alpha , alpha ^(2) , beta +gamma ),( beta , beta ^(2) , gamma +alpha ),( gamma , gamma ^(2) ,alpha +beta ) |:} =(beta -gamma ) (gamma -alpha ) (alpha -beta ) (alpha +beta +gamma )

Prove that |2 alpha+beta+gamma+delta alphabeta+gammadelta alpha+beta+gamma+delta 2(alpha+beta)(gamma+delta) alphabeta(gamma+delta)+gammadelta(alpha+beta) alphabeta+gammadeltaalphabeta(gamma+delta)+gammadelta(alpha+beta)2alphabetagammadelta|=0

Prove that |{:(betagamma,betagamma'+beta'gamma,beta'gamma'),(gammaalpha,gammaalpha'+gamma'alpha,gamma'alpha'),(alphabeta,alphabeta'+alpha'beta,alpha'beta'):}| =(alphabeta'-alpha'beta)(betagamma'-beta'gamma)(gammaalpha'-gamma'alpha) .

If alpha,beta,gamma are such that alpha+beta+gamma =4, alpha^2+beta^2+gamma^2 =6, alpha^3+beta^3+gamma^3 =8, then the value of [alpha^4+beta^4+gamma^4] must be equal to (where[.] denotes the greatest integer function)

|{:(1,alpha,alpha^3),(1,beta,beta^3),(1,gamma,gamma^3):}|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

If [[alpha, beta], [gamma, -alpha]] is to be square root of two-rowed unit matrix, then alpha,beta and gamma should satisfy the relation. a. 1-alpha^2+betagamma=0 b. alpha^2+betagamma=0 c. 1+alpha^2+betagamma=0 d. 1-alpha^2-betagamma=0

Prove that |((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|= -64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)(gamma-beta)