Question about similar matrices: if A is similar to B then what also must be similar?

Question

The answer choices are

• $A + 3I$ and $B + 3I$
• $2A$ and $2B$

I think 2 is correct based on my work:

If $A$ and $B$ are similar then $AS = SB$, $A = SBS^{-1}$, $2A = 2SBS^{-1}$, $2A = 2B$

With this result, $2A = 2B$, we know that $A = B$ so they must be similar.

The same should apply for number 1 too right?

Answer 1

You are wrong when you write that “$A=B$” and, before that, when you write that “$2A=2B$”. The conclusion is that $2A=S(2B)S^{-1}$ and that therefore $2A$ and $2B$ are similar.

The same argument applies to the other question. In fact, more generally, if $P$ is a polynomial in one variable and if $A$ and $B$ are similar, then $P(A)$ and $P(B)$ are similar. The argument is similar.