Q1: Let $A \in \mathbb{C}^{n \times n}$. What does it mean that $A$ similar to a diagonal matrix?

A1: There is an invertible matrix $X$ and a diagonal matrix $D$ such that: $$ D = X^{-1} A X $$

A2: There is a diagonal matrix $D$ such that: $$ D = A $$

A3: There is an invertible matrix $X$ such that: $$ A = X^{-1} A X $$

A4: There are $n$ eigenvectors of $A$.

Q2: Let $A \in \mathbb{C}^{n \times n}$ with $n$ eigenvectors that we put into the columns of a matrix $X$. Let $D$ be a diagonal matrix where we find the eigenvalues of $A$ on the diagonal. What is correct?

A1: $AX = XD$

A2: $A = XD$

A3: $XA = XD$

A4: $XA = DX$