If AB = BA for any two sqaure matrices, prove by mathematical ind… - Vidyadip

Question

If AB = BA for any two sqaure matrices, prove by mathematical induction that (AB)ⁿ = AⁿBⁿ.

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Answer

Let P(n) : (AB)ⁿ = AⁿBⁿ

$\therefore$ P(1) : (AB)¹ = A¹B¹ ⇒ AB = AB

So, P(1) is true.

Now, P(k) : (AB)^k = A^kB^k $\text{k}\in\text{N}$

So, P(k) is true, whenever P(k + 1) is true.

$\therefore$ P(k + 1) ⇒ (AB)^k+1 = A^k+1.B^k+1

$\therefore$ P(k + 1 : AB)^k+1 = A^k+1B^k+1

⇒ A^k.B^k.AB

⇒ A^k.B^kBA ⇒ A^kB^k+1A

⇒ A^kA.B^k+1 ⇒ A^k+1B^k+1

⇒ (A.B)^k+1 = A^k+1B^k+1

So, P(k+1) is true for all $\text{n}\in\text{N},$ whenever P(k) is true.

By mathematical induction (AB) = AⁿBⁿ is true for all $\text{n}\in\text{N}.$

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