Diagonalizability in relation to characteristic polynomial and row equivalence

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!

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 $$\begin{bmatrix}1&0\\0&1\end{bmatrix}$$.