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 $$p$$.