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Misc 12 - If AB = BA, then prove by induction that ABn = BnA - Miscellaneous

Misc. 12 - Chapter 3 Class 12 Matrices - Part 2
Misc. 12 - Chapter 3 Class 12 Matrices - Part 3 Misc. 12 - Chapter 3 Class 12 Matrices - Part 4 Misc. 12 - Chapter 3 Class 12 Matrices - Part 5 Misc. 12 - Chapter 3 Class 12 Matrices - Part 6


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Misc. 12 If A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn = Bn A .Further, prove that (AB)n = An Bn for all n ∈ N First we will prove ABn = BnA We that prove that result by mathematical induction. Step 1: Let P(n): If AB = BA , then ABn = BnA Step 2: Prove for n = 1 For n = 1 L.H.S = AB1 = AB R.H.S = B1A = BA So, L.H.S = R.H.S ∴ P(n) is true for n = 1 Step 3: Assume P(k) to be true and then prove P(k+1) is true Let us assume P(k) is true P(k): If AB = BA , then ABk = BkA We will prove that P(k+1) is true P(k+1): If AB = BA , then ABk+1 = Bk+1A Taking L.H.S ABk+1 = A.(BkB) = (ABk). B = (Bk.A)B = Bk(AB) = Bk(BA) = (BkB)A = Bk+1 A = R.H.S Hence P(k + 1) is true is when P(k) is true ∴ By the mathematical induction P(n) is true for all n Where n is natural number Hence, if AB = BA, then ABn = BnA where n ∈ N Now, we will prove that If AB = BA , then (AB)n = An Bn for all n ∈ N We shall prove the result by mathematical induction Step 1: Let P (n) : If AB = BA , then (AB)n = An Bn Step 2: Prove for n = 1 For n = 1 L.H.S = (AB)1 = AB R.H.S = A1 B1 = AB So, L.H.S = R.H.S ∴ P(n) is true for n = 1 Step 3: Assume P(k) to be true and then prove P(k+1) is true We assume that P(k) is true. P(k) : If AB = BA , then (AB)k = Ak Bk We will prove that P(k + 1)is true P (k + 1): If AB = BA , then (AB)k + 1 = Ak + 1 . Bk + 1 Taking L.H.S (AB) k + 1 = (AB) k . AB = Ak Bk (AB) = Ak Bk (BA) = Ak (Bk B)A = Ak (Bk+1) A = Ak (Bk+1 A) = Ak (A Bk+1) = (Ak A) Bk+1 = Ak+1 . Bk+1 = R.H.S Hence P(k+1) is true ∴ By the mathematical induction P (n) is true for all n (where n is natural number) Hence if AB = BA, then (AB)n = AnBn where n ∈ N

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.