Question 20
Consider two non-commuting matrices $A, B \in M_2(\mathbb{R})$.
a) If $A^3 = B^3$, then $A^n$ and $B^n$ have for any positive integer $n$ the same trace.
b) Show that there are non-commuting $A, B \in M_2(\mathbb{R})$, such that for any positive $n$, $A^n$ and $B^n$ are different.
a) If $A^3 = B^3$, then $A^n$ and $B^n$ have for any positive integer $n$ the same trace.
b) Show that there are non-commuting $A, B \in M_2(\mathbb{R})$, such that for any positive $n$, $A^n$ and $B^n$ are different.