Math GRE - #15

Suppose $A$ and $B$ are invertible $n\times n$ matrices. If $A$ and $B$ are similar, which of the following statements are true?

1. $A-2I$ and $B-2I$ are similar.
2. $A$ and $B$ have the same trace.
3. $A^{-1}$ and $B^{-1}$ are similar matrices.

Choices:

• 1 only.
• 2 only.
• 3 only.
• 1 and 2 only.
• 1, 2, and 3 only.

Solution :

1, 2, and 3 are all true.

Recall that $A$ and $B$ are similar if $B=P^{-1}AP$ for some invertible matrix $P$.

 

Since $\text{tr}(AB)=\text{tr}(BA)$, we know that 2 is true due to: \[\text{tr}(B)=\text{tr}(P^{-1}AP)=\text{tr}(APP^{-1})=\text{tr}(A).\]

Moreover, we have \[B^{-1}=(P^{-1}AP)^{-1}=P^{-1}A^{-1}P.\] But by the definition of being similar, this means that $A^{-1}$ and $B^{-1}$ are similar. So 3 is true. Since only the last choice offers 2 and 3 as true, we know that the answer is the last choice: 1, 2, and 3 are all true.

Can you prove why 1 must be true?