A = cosX sinX

-sinX cosX

then prove that

Aⁿ = cos nX sin nX

-sin nX cos nX , n belongs to all natural no.

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

Now we will show that the result is true for n = m + 1 i.e.,

By the definition of integral powers of a square matrix, we have

This shows that the result is true for n = m + 1, whenever it is true for n = m.

Hence, by the principle of mathematical induction the result is valid for any positive integer n.