If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$, show by induction that $A^{n}=\left[\begin{array}{cc}\cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta\end{array}\right]$.

CBSE Class 12 · 2006 · MatricesLong Answer

A = [cos θ - sin θ sin θ cos θ]

, show by induction that

A^{n} = [cos n θ - sin n θ sin n θ cos n θ]

Answer: Proved

Base case

n = 1

is the given matrix, so the statement holds.
Inductive step: assume

A^{k} = [cos k θ - sin k θ sin k θ cos k θ]

. Then

A^{k + 1} = A^{k} A = [cos k θ - sin k θ sin k θ cos k θ] [cos θ - sin θ sin θ cos θ] .

Using

cos k θ cos θ - sin k θ sin θ = cos (k + 1) θ

and

cos k θ sin θ + sin k θ cos θ = sin (k + 1) θ

A^{k + 1} = [cos (k + 1) θ - sin (k + 1) θ sin (k + 1) θ cos (k + 1) θ] .

By the principle of mathematical induction the result holds for all

n \in N

. Proved.

CBSE Class 12 Board · 2006 · Subjective