Prove that `|{:(alpha,beta,gamma),(theta,phi,psi),(lambda,mu,nu):}|=|{:(beta,mu,phi),(alpha,lambda,theta),(gamma,nu,psi):}|.`

Using peoperties of determinants in questions 11 to 15, prove that : |{:(alpha,alpha^(2),beta+gamma),(beta,beta^(2),gamma+alpha),(gamma,gamma^(2),alpha+beta):}|=(beta-gamma)(gamma-alpha)(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),2alphabetagamma delta):}| =0

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

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

If alpha,beta "and" gamma are real number without expanding at any stage prove that |{:(1,cos(beta-alpha),cos(gamma-alpha)),(cos(alpha-beta),1,cos(gamma-beta)),(cos(alpha-gamma),cos(beta-gamma),1):}| =0.

Using properties of determinants, prove the following |(alpha,beta,gamma),(alpha^2,beta^2,gamma^2),(beta+gamma,gamma+alpha,alpha+beta)|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

Show that |(1,alpha,alpha^3),(1,beta,beta^3),(1,gamma,gamma^3)|=(alpha-beta)(beta-gamma)(gamma-alpha)( alpha+beta+gamma)

Determine the values of alpha ,beta , gamma when [{:( 0, 2beta, gamma),( alpha ,beta, -gamma),( alpha ,-beta, gamma):}] is orthogonal.