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Let A= [[1,4],[3,2]] If theta is the ang...

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

A

3

B

5

C

7

D

9

Text Solution

Verified by Experts

The correct Answer is:
C
`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`
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