Diagonalizability in relation to characteristic polynomial and row equivalence

Question

I am new to linear algebra, and am unsure re the following question:

True or False?

Let A and B be matrices of n x n.

1. If A and B are diagonalizable and they have the same characteristic polynomial, then A and B are similar.

2. If A and B are row equivalent and A is diagonalizable, then B is diagonalizable.

My intuitive answer is "false" to 1, and "true" to 2.

However, I am not sure, and either way, I would ideally like to be able to prove it...

Many thanks!

Answer 1

Wrong on both counts.

For (1): If $A$ is diagonalizable then $A$ is similar to $D$, where $D$ is a diagonal matrix that has the eigenvalues of $A$ on the diagonal. If $A$ and $B$ have the same characteristic polynomial then they have the same eigenvalues, so if $B$ is diagonalizable it's similar to the same $D$ as we used for $A$. So $A$ and $B$ are similar.

For (2): $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ is row-equivalent to $\begin{bmatrix}1&0\\0&1\end{bmatrix}$.