Are products and sums of similar matrices similar?

Question

Assume that $A$ and $B$ are similar matrices. Which of the following is true?

a) Matrices $AB$ and $BA$ can't be similar.

b) Matrices $A + B$ and $B + A$ can't be similar.

c) Matrices $AA$ and $B$ can't be similar.

d) None of the above.

I have a problem with the simillarity definition. I know that $A$ and $B$ are similar if there exists a matrix $P$ such that $A = P^{-1}BP$. Does that, however mean that $B = P^{-1}AP$ with the same matrix $P$? If not, than:

Let $A = P^{-1}BP$ and $B = R^{-1}AR$. Then:

a) $AB = P^{-1}BPR^{-1}AR$ which tells me nothing.

How should I approach this type of problem?

Answer 1

The example $A=B=I$ shows that a), b) and c) are all false.

Answer 2

Kavi Rama Murthy already answered your first question. For your second question: if $A = P^{-1} B P$, then multiplying by $P$ on the left and multiplying by $P^{-1}$ on the right gives $B=PAP^{-1} = (P^{-1})^{-1}A P^{-1}$.

Answer 3

In linear algebra, two n-by-n matrices $A$ and $B$ are called similar if there exists an invertible n-by-n matrix $P$ such that $B=P^{-1}AB$.

According to this relation, we can assume that similarity is an equivalence relation on the space of square matrices.

This equivalence relation pretty much says that $A=B=I$ (as Kavi Rama Murthy said). So we can assume that $AB=BA$ since it's the same as $A^2=A^2$ (or $B^2=B^2$). Using the same equality, the second question is obviously false as Lucas L. showed using $P=(P^{-1})^{-1}$. Finally, the third one might be the easiest one to understand: $A=B$, but with the relation above, they are also equal to $I$, the identity matrix which has the following property: $$I^2=I$$

$$\Leftrightarrow AA=A^2=I=B$$

The answer to your exercise is d) because as we demonstrated it, a), b) and c) are all false.