Problem 6

 

Let and (the determinant of T). Furthermore, let a+d (the trace of T) equal for some angle theta. Finally, let .  Show that for all natural numbers q, where I is the identity matrix.

 

 

Solution:

The usual way to prove such a formula is by using the principle of mathematical induction.  Since the formula is trivially true when , all we need to do is prove that is we assume the formula is true for q (the induction hypothesis) , then it is also true for q+1.  I.e.  we need to prove that .  We begin:

So, we will be done if we can show that .  But this is equivalent to .  Applying the appropriate “product to sum” trigonometric identity implies the result.