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)`

Prove that : |[alpha+beta+gamma,-gamma,-beta],[-gamma, alpha+beta+gamma, -alpha],[-beta,-alpha,alpha+beta+gamma]|= 2(alpha+beta)(beta+gamma)(gamma+alpha)

Using properties of determinants, 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+gammadelta,,alphabeta(gamma+delta)+gammadelta(alpha+beta),,2alphabetagammadelta):}|=0

Prove that cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4cos(alpha+beta)/(2)cos(beta+gamma)/2cos(gamma+alpha)/2

if alpha,beta,gamma are the roots of x^3-3x^2 +3x + 7 =0 then (alpha-1)/(beta-1)+(beta-1)/(gamma-1)+(gamma-1)/(alpha-1)

If [(0,2beta,gamma),(alpha,beta,-gamma),(alpha,-beta,gamma)] is orthogonal, then find the value of 2alpha^(2)+6beta^(2)+3gamma^(2).

If alpha, beta, gamma are the roots of the cubic x^(3)-px^(2)+qx-r=0 Find the equations whose roots are (i) beta gamma +1/(alpha), gamma alpha+1/(beta), alpha beta+1/(gamma) (ii) (beta+gamma-alpha),(gamma+alpha-beta),(alpha+beta-gamma) Also find the valueof (beta+gamma-alpha)(gamma+alpha-beta)(alpha+beta-gamma)

Without expanding, prove the following |(1,1,1),(alpha,beta,gamma),(betagamma,gammaalpha,alphabeta)|=(beta-alpha)(gamma-alpha)(alpha-beta)

In the fig. Prove that x=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