Question

Find the eigenvalues with corresponding eigenvectors & show work.

1/2 1/9 3/10

1/3 1/2 1/5

1/6 7/18 1/2

Answer 1

Let

Let be the eigenvalue of the matrix A.

The from the definition of eigenvalue we have,

[I is the identity matrix of order 3]

By expanding we get,

[assuming ]

540a3 + 270a2 - 270a2 - 135a + 46a + 23 = 0

270a2(2a + 1) - 135a(2a + 1) + 23(2a + 1) = 0

(2a + 1)(270a2 - 135a + 23) = 0

Either a = - 1/2

Or, 270a2 - 135a + 23 = 0

i.e.

Since,

Therefore, there there is no real value of a

Therefore a = - 1/2 and hence = 1 i.e. the eigenvalue of A is 1.

Now, we find the engenvector corresponding to this eigenvalue. Let v be the eigenvector.

Then, Av = v

1/2.x + 1/9.y + 3/10.z = x

and 1/3.x + 1/2.y + 1/5.z = y

and 1/6.x + 7/18.y + 1/2.z = z

45x - 10y - 9z = 0

and 10x - 15y + 6z = 0

and 3x + 7y - 9z = 0

Solving we get x = y = z = 0

Answer: The eigenvalue for the given matrix is 1 and the eigenvector is .