Option B is the correct answer.

Given \( \rm \left ( x + \frac{1}{x} = 2\right ) \)

We can work the equation to receive:

\( \rm x^{2} + 1 = 2x \)

\( \rm x^{2} - 2x + 1 = 0 \)

\( \rm (x - 1)^{2} = 0 \)

The above is a quadratic equation in x. We can apply the quadratic formula to solve for x. Or, we can solve it as follows:

\( \rm (x - 1) = \sqrt{0} \)

\( \rm (x - 1) = 0 \)

\( \rm x = 1 \)

Hence, the only value of x that satisfies the given equation is 1.

Again, substitute x = 1 in the second equation to find its value.

\( \rm  x^{100} + \frac{1}{x^{100}} \)

\( \rm = 1^{100} + \frac{1}{1^{100}} \)

\( \rm = 1 + \frac{1}{1} \)

\( \rm = 2 \)

Hence, the required value of the expression is 2.