If the matrix `A=[[1,0,0] , [0,2,0], [3,0,-1]]` satisfies the equation `A^(20)+alpha A^(19)+beta A = [[1,0,0] , [0,4,0] , [0,0,1]]` for some real numbers `alpha` and `beta` then `beta-alpha=`

If [[1,0,0],[0,2,0],[3,0,-1]] and A^20 +alphaA^19+betaA=[[1,0,0],[0,4,0],[0,0,1]] Find the value of (alpha+beta)

If A=[[0,1] , [-1,0]]=(alphaI+betaA)^2 then alpha and beta =

Let A = [(3,-1),(0,2)] . Suppose A satisfies the equation x^(2)+ax+b=0 for some real numbers a and b . Let alpha , beta be the roots of t^(2)+at +b=0 then (1)/(alpha^(2)-3alpha+4)+(1)/(beta^(2)-3alpha+4) = ________

If coa(alpha+beta)+sin(alpha-beta)=0 and 2010tan beta+1=0 then tan alpha=

If the roots alpha and beta of the equation , x^(2) - sqrt(2) x + c = 0 are complex for some real number c ne 1 and |(alpha - beta)/( 1 - alpha beta)| = 1, then a value of c is :

A=([1,0,0],[0,1,1],[0,-2,4])" and if "A^(-1)=(1)/(6)[alpha A^(2)+beta A+gamma I]" when "I" is unit matrix of order "3" then the value of "alpha+beta+gamma" is "

If alpha, beta are roots of the equation x^(2) + x + 1 = 0 , then the equation whose roots are (alpha)/(beta) and (beta)/(alpha) , is

If sin alpha sin beta-cos alpha cos beta+1=0, then prove that 1+cot alpha tan beta=0