[" 44"A=[[alpha,beta],[gamma,-alpha]]" st that "],[[" (A) "1+alpha^(2)+beta y=0]]

If alpha, beta, gamma are the roots of x^3+a x^2+b=0 then the value of [[alpha , beta , gamma],[beta , gamma , alpha],[gamma , alpha , beta]] is

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 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 .

Evaluate the following: |[1,1,1],[alpha, beta, gamma],[beta gamma, gamma alpha, alpha beta]|

Prove that |[[alpha,beta,gamma],[alpha^2,beta^2,gamma^2],[alpha^3,beta^3,gamma^3]]|= alpha beta gamma (alpha-beta)(beta-gamma)(gamma-alpha)

If alpha, beta, gamma are the roots of ax ^ (3) + bx ^ (2) + cx + d = 0 and det [[alpha, beta, gammabeta, gamma, alphagamma, alpha, beta]] = 0, alpha ! = beta! = gamma, alpha + beta-gamma, beta + gamma-alpha, and gamma + alpha-beta

If alpha, beta, gamma be the zeros of the polynomial p(x) such that (alpha+beta+gamma) = 3, (alpha beta+beta gamma+gamma alpha) =- 10 and alpha beta gamma =- 24 then p(x) = ?

Without expanding the determinant, prove that: (i) |{:(alpha, alpha^(2), beta gamma),(beta, beta^(2), gamma alpha),(gamma, gamma^(2), alpha beta):}| =|{:(1,alpha^(2), alpha^(3)),(1, beta^(2), beta^(3)),(1, gamma^(2), gamma^(3)):}|