Question

If `A=[[cos θ, i sinθ],[i sinθ,cosθ]]` then prove by principle of mathematical induction that `A^n=[[cos  nθ,i sinθ],[i sin nθ,cos nθ]]` for all `n  ∈ N.`

Answer 1

We shall prove the result by the principle of mathematical induction on n.

Step 1: If n = 1, by definition of integral power of a matrix, we have

\[A^1 = \begin{bmatrix}\cos 1\theta & i \sin1\theta \\ i \sin 1\theta & \cos 1\theta\end{bmatrix} = \begin{bmatrix}\cos\theta & i \sin\theta \\ i \sin\theta & \cos\theta\end{bmatrix} = A\]

Thus, the result is true for n=1.

Step 2: Let the result be true for n = m. Then,

`A^m = [[ cos mθ        i sin mθ ],[i sin mθ     cos mθ ]]`

Now we shall show that the result is true for

`n=m+1`

Here

\[A^{m + 1} = \begin{bmatrix}\ cos \left( m + 1 \right)\theta & i \ sin\left( m + 1 \right)\theta \\ i  \ sin \left( m + 1 \right)\theta & \ cos \left( m + 1 \right)\theta\end{bmatrix}\]            ...(1)

By definition of integral power of matrix, we have

\[A^{m + 1} = A^m A\]

`⇒ A^m+1 = [[ cos m θ    i sin m θ],[ i sin m θ     cos m θ  ]]` `[[cos θ   i sin θ  ],[ i sin θ   cos θ  ]]` [From  eq (1) ]

`⇒ A^m+1 = [[ cos m θ  .cos θ+  i sin m θ.i sin θ     cos m θ  . i sin θ  +  i sin m θ . cos θ ],[  i sin m  θ  .  cos θ  + cos m  θ . i sin   θ    i sin m θ . i sin   θ  + cos m  θ .cos  θ  ]]`

`⇒ A^m+1 = [[ cos m θ  .cos θ-  sin m θ. sin θ    i  ( cos m θ  .  sin θ  +  sin m θ . cos θ )],[  i (sin m  θ  .  cos θ  + cos m  θ . i sin   θ  ) -  sin m θ . i sin   θ  + cos m  θ .cos  θ  ]]`

`⇒ A^m+1 = [[ cos m θ  .cos θ-  sin m θ. sin θ    i  ( cos m θ  .  sin θ  +  sin m θ . cos θ )],[  i (sin m  θ  .  cos θ  + cos m  θ .  sin   θ  )      cos m  θ .cos θ  -  sin m θ . sin θ  ]]`

`⇒ A^m+1 =[[ cos (m θ + θ )   i sin (m θ + θ ) ], [ i sin (m θ + θ )     cos (m θ + θ )]]`

`⇒ A^m+1 =[[ cos (m  + 1)θ  i sin (m  +1) θ ], [ i sin (m  + 1 )θ    cos (m  + 1)θ ]]`

 

This shows that when the result is true for n = m, it is true for

`n=m+1`

Hence, by the principle of mathematical induction, the result is valid for all n

\[\in N\]

Disclaimer: n is missing before

\[\theta\] in a₁₂ in Aⁿ.