A15 10 10 10 eigenvalues and eigenvecorsSolutionSolution of

A=(15 -10

10 -10)

eigenvalues and eigenvecors?

Solution of A - lambda*I = 0, gives the eigen value

= (15 -10) - A(1 0) = 0

(10 -10) - A(0 1) = 0

= (15 - A, -10) = 0

(10, (-10 - A)) = 0

= -(15 - A)*(10 + A) - 10*(-10) = 0

= A^2 --5A -150 + 100 = 0

= A^ - 5A - 50 = 0

Solving this equation

A = [5 +/- sqrt (5^2 - 4*1*(-50))]/2

A = [5 + sqrt 225]/2 and [5 - sqrt 225]/2

Wigen Values will be A = 10 and -5

So solution will be

A - 10I and A + 5I

Now for eiganvectors

(A - 10I)*x = 0

(5 -10)*(x1) = 0

(10 -20)*(x2) = 0

5*x1 -10*x2 = 0

x1 = -2*x2

10x1 - 20x2 = 0

x1 = -2*x2

eigenvector1 will be

(x1 x2) = k*(2, -1)

Similarly

A + 5I = 0

(20 -10)(x3) = 0

(10 -5)(x4) = 0

20x3 - 10x4 = 0

x4 = 2*x3

10x3 - 5x4 = 0

x4 = 2*x3

So eigenvector2 will be

(x3, x4) = k*(1, 2)

where k is a random constant.

Comment below if you have any doubt.