# Assertions about a square, diagonalizable matrix

How many of the following assertions are true for a $$n \times n$$ diagonalizable matrix $$X$$?

• There is an invertible matrix $$Q$$ such that $$QXQ$$-1

I think this assertion is true because if $$X$$ is diagonalizable, then there exists an inverttible matrix P such that $$P$$-1$$XP$$ is a diagonal matrix. Let $$Q$$=$$P$$-1 then we have that $$QXQ$$-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$$XP$$ 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.

• 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.