Assertions about a square, diagonalizable matrix

Question

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.

Answer 1

You are correct!

• Assuming you meant ... "is diagonal", then yes, since $\mathbf P$ is invertible $\mathbf Q = \mathbf P^{-1}$ works.
• Again, this is true. If you like, you can justify it further by writing $\mathbf X = \mathbf P \mathbf D \mathbf P^{-1}$ for some diagonal matrix $\mathbf D$, and then each column of $\mathbf P$ must be an eigenvector of $\mathbf X$, but since $\mathbf P$ is invertible, its columns must be linearly independent.
• Once again, you're correct. When the answer to a question like this is negative, it's often a good idea to provide an explicit counterexample if you can find one. This can be helpful to avoid certain kinds of flawed reasoning. In this case, you could use \begin{equation*} \mathbf I^{-1} \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \mathbf I = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \end{equation*} which is clearly not invertible.