Section B question 1

Question

Answer 1

Let P(n) : IfA = \(\begin{bmatrix}cos\theta&sin\theta\\-sin\theta&cos\theta\end{bmatrix}\) then Aⁿ = \(\begin{bmatrix}cosn\theta&sinn\theta\\-sinn\theta&cosn\theta\end{bmatrix}\) 

P(1) is true

\(\therefore\) A² = \(\begin{bmatrix}cos^2\theta-sin^2\theta&2sin\theta cos2\theta\\-2sin\theta cos2\theta&cos^2\theta-sin^2\theta\end{bmatrix}\)

\(=\begin{bmatrix}cos2\theta&sin2\theta\\-sin2\theta&cos2\theta\end{bmatrix}\)

P(2) is true

\(\therefore\) A³  = \(=\begin{bmatrix}cos\theta cos2\theta-sin\theta sin2\theta&sin2\theta cos\theta+cos2\theta sin\theta\\-(sin2\theta cos\theta+sin\theta cos2 \theta)&cos \theta cos2\theta-sin\theta sin2\theta\end{bmatrix}\)

= \(\begin{bmatrix}cos3\theta&sin3\theta\\-sin3\theta&cos3\theta\end{bmatrix}\)

\(\therefore\) P(3) is true

Let P(n) is true

Hence, Aⁿ = \(\begin{bmatrix}cosn\theta&sinn\theta\\-sinn\theta&cosn\theta \end{bmatrix}\)

Now, we have to chek P(n + 1) is true or not.

Aⁿ⁺¹ = Aⁿ.A  = \(\begin{bmatrix}cosn\theta&sinn\theta\\-sinn\theta&cosn\theta \end{bmatrix}\) \(\begin{bmatrix}cos\theta&sin\theta\\-sin\theta&cos\theta \end{bmatrix}\) 

= \(\begin{bmatrix}cos n \theta cos\theta sinn\theta sin \theta & sin\theta cos n\theta+cos\theta sin n\theta\\-(sin\theta cosn\theta+cos\theta sin n\theta) & cos n\theta cos\theta-sin n\theta sin\theta\end{bmatrix}\) 

= \(\begin{bmatrix}cos(n+1)\theta&sin(n+1)\theta\\-sin(n+1)\theta&cos(n+1)\theta\end{bmatrix}\)

Hence, P(n + 1) is true.

∴ P(n) is true for all n \(\in \) N.

Hence, If A = \(\begin{bmatrix}cos\theta&sin\theta\\-sin\theta&cos\theta \end{bmatrix}\) then Aⁿ = \(\begin{bmatrix}cosn\theta&sinn\theta\\-sinn\theta&cosn\theta \end{bmatrix}\)