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.