According to the principle of mathematical induction, if a statement “S(n) is true
for n = 1” (or for some starting point j), and if “S(n) is true for n = k” implies that “S(n)
is true for n = k + 1” (whatever integer k ≥  j may be), then the statement is true for any
positive integer n, for all n ≥ j We now consider some examples Example 3 Show that if
 A = 
cos
sin
sin
cos
θ
θ




−
θ
θ

 , then   An =
cos
sin
sin
cos
n
n
n
n
θ
θ




−
θ
θ


Solution We have
P(n) : An =
cos
sin
sin
cos
n
n
n
n
θ
θ




−
θ
θ


We note that
P(1) : A1 =
cos
sin
sin
cos
θ
θ




−
θ
θ


Therefore, P(1) is true Assume that P(k) is true, i