If UX = XI, then x11 = x22 =0, x21 =0, x31 = x42 = 0 and x41 = 0. Thus, the first column of X is all zeros and X is not invertible.
If U were similar to I, then here would be an invertible matrix X that fulfills I = X− 1UX, and so XI = UI. Thus, it proves that there can be no such invertible matrix X. Hence, U is not similar to I.
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