Matrix similarity proof

Question

I need to show that the matrices

$$ \begin{pmatrix} 1 & 1 \\ 0 & 0 \\ \end{pmatrix} $$ and $$ \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} $$

aren't similar. I tried with trace, determinant, eigenvalues, rank space but their all the same.

Can someone help?

Answer 1

That will be hard, since they are similar:$$\begin{bmatrix}1&1\\0&-1\end{bmatrix}^{-1}.\begin{bmatrix}1&0\\0&0\end{bmatrix}.\begin{bmatrix}1&1\\0&-1\end{bmatrix}=\begin{bmatrix}1&1\\0&0\end{bmatrix}.$$

Answer 2

The "softest" way to see they're similar: Being upper triangular, they both have eigenvalues $0$ and $1$. Hence, being $2\times 2$, they're both diagonalizable, and with the same diagonal matrix.