eigen vector

Question

An arbitrary vector X is an eigen vector of the vector of the matrix A=[1 0 0    a 0 0    0 0 b], if (a, b)=
(a) (0, 0) (b) (1, 1)
(c) (0, 1) (d) (1, 2)

Answer 1

Given Matrix is A = [1 0 0 ; a 0 0 ; 0 0 b]

Since it is a upper triangular matrix ; eigen values will be the diagonal elements (1, a , b);

Now suppose 1 as the eigen value(c) of the arbitrary vector ; then we have the equation AX = cX => AX = 1X

(1 0 0 ; a 0 0; 0 0 b) (X1; X2; X3) = 1(X1; X2; X3)

Solving the above; u will get the following equations

X1 = X1 ; aX1 = X1 ; bX3 = X3

So bX3 = X3 => b =1 -----> (1)

     aX1 = X1 => a=1

So (a,b) = (1,1) is the answer.