if `A=[[sin alpha, cos alpha],[-cos alpha, sin alpha]]`, then verify `A^T A=I`

If A=[[cos alpha, -sin alpha],[sin alpha, cos alpha]] , then A+A^T=I if the value of alpha is

If A_(alpha)=[[cos alpha, sin alpha],[ -sin alpha , cos alpha]] , then prove that i) A_(alpha ). A_(beta)=A_(alpha + beta) ii) (A_alpha)^n=[[cos n alpha, sin n alpha],[ -sin n alpha, cos n alpha]] for every positive integer n .

Evaluate Delta=|[0, sin alpha, -cos alpha],[ -sin alpha, 0 , sin beta],[ cos alpha, -sin beta, 0]|

If A = [(cos alpha,sin alpha),(-sin alpha,cos alpha)] then A^(10) is equal to

Find inverse of the matrix A=[[cos alpha, -sin alpha, 0 ],[ sin alpha, cos alpha, 0],[ 0, 0, 1]]

Evaluate |[cos alpha cos beta, cos alpha sin beta, -sin alpha],[ -sin beta, cos beta, 0],[ sin alpha cos beta, sin alpha sin beta, cos alpha]|

Let lambda and alpha be real. Let S denote the set of all values of lambda for which the system of linear equations lambda x+ (sin alpha) y + (cos alpha)z = 0 , x + (cos alpha) y + (sin alpha) z = 0 , -x + (sin alpha) y - (cos alpha) z = 0 has a non trivial solution then S contains a) (-1,1) b) [-sqrt2,-1] c) [1,sqrt2] d) [-sqrt2,sqrt2]

If f(alpha,beta)=|(cosalpha,-sinalpha,1),(sin alpha,cosalpha,1),(cos(alpha+beta),-sin(alpha+beta),1)| , then a) f(300,200) ne f(400, 200) b)f(200,400) = f(200,600) c)f(100,200) =f(200,200) d)None of these

If A_(alpha)=[(cos alpha,sinalpha),(-sinalpha,cosalpha)] then A_(alpha) A_(beta) is equal to A) A_(alpha beta) B) A_(alpha +beta) C) A_(alpha - beta) D)None of these

If |(sin^(2)alpha,cos^(2)alpha),(cos^(2)alpha,sin^(2)alpha)|=0,alphain(0,pi) , then the values of alpha are