showing that 2 matrices are not similar

Question

There are two $3\times 3$ matrices: $$ A = \begin{bmatrix} 2 &-1 &-1\\ 0& 1 &1\\ 0 &0 &2 \end{bmatrix} $$ $$ B = \begin{bmatrix} 2 &-1 &1\\ 0& 1 &1\\ 0& 0& 2 \end{bmatrix} $$ I need to show that these are not similar. They have the same determinant, rank and trace. I've tried to subtract with a matrix of the form $xI$ so that $x$ is a real number but that didn't work. Thanks in advance!

Answer 1

Hint. Look at the rank of $A - 2I$ and $B- 2I$.

Answer 2

this is what we get by row reducing the two matrices $A-2I = \begin{bmatrix} 0 &-1 &-1\\ 0& -1 &1\\ 0 &0 &0 \end{bmatrix} \to \begin{bmatrix} 0 &1 &0\\ 0& 0 &1\\ 0 &0 &0 \end{bmatrix}$ and

$B-2I = \begin{bmatrix} 0 &-1 &1\\ 0& -1 &1\\ 0 &0 &0 \end{bmatrix} \to \begin{bmatrix} 0 &1 &-1\\ 0& 0 &0\\ 0 &0 &0 \end{bmatrix}$

so that the null space of $A-2I$ has dimension one and the null space of $B-2I$ has dimension two. that is the matrices $A$ and $B$ are not similar.