True and False. Linear Algebra question

Question

My solutions are as follows

1: True 2: False 3: False 4: True 5: True

Are these solutions reasonable? Thank you

Answer 1

The second one is true, the rest of your solutions is correct.

As for the second one: similarity of $A$ and $B$ means that for some invertible $S$ we have $B= SAS^{-1}$. (Note that $A$ and $B$ have the same type and they are square matrices as they are invertible, so all three of the matrices $A,B,S$ are $n\times n$ for some $n$.)

Take the inverse of both sides (makes sense, as the product of invertible matrices is invertible): $B^{-1}= SA^{-1}S^{-1}$, so $A^{-1}$ and $B^{-1}$ are also similar. In fact, you can conjugate by the same matrix that witnesses the similarity of $A$ and $B$.