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 \(p\).

\(a\):
\(b\):
\(c\):
\(d\):

\(e\):
\(f\):
\(g\):
\(h\):

\(p\):