If `A= [[3,2,2],[2,4,1],[-2,-4,-1]] ` and X,Y are two non-zero
column vectors such that `AX=lambda X, AY=muY , lambdanemu, ` find
angle between X and Y.

Explanation

`because AX = lambda X rArr (A-lambdaI) X = 0`
`because = X ne 0`
`therefore det (A-lambdaI)=0`
`rArr [[3-lambda,2,2],[2, 4-lambda,1],[-2, -4,-1-lambda]]=0`
Applying `R-(3) rarr R_(3) + R_(2)`, then
` [[3-lambda,2,2],[2, 4-lambda,1],[0, -lambda,-lambda]]=0`
Applying `C_(2 ) rarr C_(2) C_(3),` then
`rArr [[3-lambda,0,2],[2, 3-lambda,1],[0, 0,-lambda]]=0`
`rArr -lambda (3-lambda )^(2) = 0`
` rArr lambda = 0,3`
It is clear that `lambda = 0 mu = 3`
for `lambda = 0, AX = 0 rArr [[3,2,2],[2,4,1],[-2,-4,-1]][[x],[y],[z]] = [[0],[0],[0]]`
`rArr 3x+ 2y+ 2z = 0 and 2x + 4y + z=0`
`therefore x/-5= y/1 = z/8`
So, `X= [[-6],[1],[8]]`
For` mu= 3, (A-3I) Y=0`
` rArr [[0,2,2],[2,1,1],[-2,-4,-4]][[alpha],[beta],[gamma]] = [[0],[0],[0]]`
`rArr 0.alpha +2beta + 2gamma = 0and 2 alpha + beta + gamma = 0`
`therefore alpha/0 = beta / 4 = gamma/(-4)`
`rArr alpha/0 = beta / -1= gamma/1`
So, `Y=[[0],[-1],[1]]`
If `theta` angle between X and Y, then
`cos theta = (0.(-6)+(-1)cdot 1 + 1cdot8)/(sqrt((0+1+1))sqrt(36+1 +64) ) = 7/sqrt(202)`