How many of the following assertions are true for a $n \times n$ diagonalizable matrix $X$?
- There is an invertible matrix $Q$ such that $Q$$X$$Q$-1
I think this assertion is true because if $X$ is diagonalizable, then there exists an inverttible matrix P such that $P$-1$X$$P$ is a diagonal matrix. Let $Q$=$P$-1 then we have that $Q$$X$$Q$-1 is a diagonal matrix.
- $X$ has n linearly independent eigenvectors
This assertion is true because a diagonalizable matrix has $n$ linearly independent vectors.
- $X$ is invertible
I don't think this one is true because if $X$ has $0$ as an eigenvalue, $D$=$P$-1$X$$P$ will not be invertible and since $X$ and $D$ are similar, X will not be invertible.
This is from a past paper question for Linear Algebra, it says that the answer is 2 of the assertions are true but it does not say which of them are true.