Let `A= [[1,4],[3,2]]` If `theta` is the angle between the two non- zero column vectors X such that `AX = lambda X` for some scalar `lambda` then `tan theta` is equal to

Options

A. 3
B. 5
C. 7
D. 9

Answer

C

Explanation

`because (A - lambdaI)X = 0`
`therefore abs(A - lambdaI) = 0`
`rArr abs((1-lambda, 4),(3, 2-lambda))= 0 `
`rArr lambda^(2) - 3 lambda - 10 = 0`
`therefore lambda = -2, 5`
For `lambda = -2 rArr [[x],[y]]= [[4],[-3]]`
For `lambda = 5rArr [[x],[y]]= [[1],[1]]`
`therefore cos theta = (4.1 + (-3) cdot1)/(sqrt((16 +9)) sqrt((1+1) ))=1/(5sqrt(2))`
`therefore tan theta = sqrt((sec^(2)theta - 1))= sqrt(49) = 7`