Exercise 5. This software computes the order of elements in the $GL(2,Z_p)$. Enter several choices for matrices $A$ and $B$. The software returns $|A|$,$|B|$, $|AB|$, $|BA|$, $|A^{-1}BA|$ and $|B^{-1}AB|$. Do you see and relationship between $|A|$,$|B|$, and $|AB|$? Do you see any relationship between $|AB|$ and $|BA|$? Make a conjecture about this relationship. Test your conjecture for several other choices for $A$ and $B$. Do you see any relationship between $|B|$ and $|A^{-1}BA|$? Do you see any relationship between $|A|$ and $|B^{-1}AB|$? Make a conjecture about this relationship. Test your conjecture for several other choices for $A$ and $B$.

Please enter the entries of $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $B = \begin{bmatrix} e & f \\ g & h \end{bmatrix}$, and a prime $p$.

$a$:
$b$:
$c$:
$d$:

$e$:
$f$:
$g$:
$h$:

$p$: