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Misc 2 - If A = [1 1 1 1 1 1], prove An = [3n-1 3n-1 3n-1 - Proof using mathematical induction

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

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Misc. 2 If A = [ 8(1&1&1@1&1&1@1&1&1)] , prove that An = [ 8(3^( 1)&3^( 1)&3^( 1)@3^( 1)&3^( 1)&3^( 1)@3^( 1)&3^( 1)&3^( 1) )], n N We shall prove the result by using mathematical induction Step 1: P(n) : If A = [ 8(1&1&1@1&1&1@1&1&1)] , An = [ 8(3^( 1)&3^( 1)&3^( 1)@3^( 1)&3^( 1)&3^( 1)@3^( 1)&3^( 1)&3^( 1) )] Step 2: Prove for n = 1 For n = 1 L.H.S = A1 = A = [ 8(1&1&1@1&1&1@1&1&1)] R.H.S = [ 8(3^(1 1)&3^(1 1)&3^(1 1)@3^(1 1)&3^(1 1)&3^(1 1)@3^(1 1)&3^(1 1)&3^(1 1) )] = [ 8(30&30&30@30&30&30@30&30&30)] = [ 8(1&1&1@1&1&1@1&1&1)] 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 Assuming P(k) is true P(k): If A = [ 8(1&1&1@1&1&1@1&1&1)] , Ak = [ 8(3^(k 1)&3^(k 1)&3^(k 1)@3^(k 1)&3^(k 1)&3^(k 1)@3^(k 1)&3^(k 1)&3^(k 1) )] We will prove that P (k + 1)is true P (k + 1): If A = [ 8(1&1&1@1&1&1@1&1&1)] , then Ak+1 = [ 8(3^((k+1) 1)&3^((k+1) 1)&3^((k+1) 1)@3^((k+1) 1)&3^((k+1) 1)&3^((k+1) 1)@3^((k+1) 1)&3^((k+1) 1)&3^((k+1) 1) )] Ak+1 = [ 8(3^(k+1 1)&3^(k+1 1)&3^(k+1 1)@3^(k+1 1)&3^(k+1 1)&3^(k+1 1)@3^(k+1 1)&3^(k+1 1)&3^(k+1 1) )] Consider L.H.S Ak +1 = Ak . A1 =[ 8(3^(k 1)&3^(k 1)&3^(k 1)@3^(k 1)&3^(k 1)&3^(k 1)@3^(k 1)&3^(k 1)&3^(k 1) )][ 8(1&1&1@1&1&1@1&1&1)] = [ 8(3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1)&3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1)&3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1)@3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1)&3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1)&3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1)@3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1)&3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1)&3^(k 1) (1)+3^(k 1) (1)+3^(k 1) (1) )] = [ 8(3^(k 1) +3 ^(k 1)+3^(k 1)&3^(k 1) +3 ^(k 1)+3^(k 1)&3^(k 1) +3 ^(k 1)+3^(k 1)@3^(k 1) +3 ^(k 1)+3^(k 1)&3^(k 1) +3 ^(k 1)+3^(k 1)&3^(k 1) +3 ^(k 1)+3^(k 1)@3^(k 1) +3 ^(k 1)+3^(k 1)&3^(k 1) +3 ^(k 1)+3^(k 1)&3^(k 1) +3 ^(k 1)+3^(k 1) )] = [ 8( 3.3 ^( 1)& 3.3 ^( 1)& 3.3 ^( 1)@ 3.3 ^( 1)& 3.3 ^( 1)& 3.3 ^( 1)@ 3.3 ^( 1)& 3.3 ^( 1)& 3.3 ^( 1) )]= [ 8(3^k&3^k&3^k@3^k&3^k&3^k@3^k&3^k&3^k )] = 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 Thus if A = = [ 8(1&1&1@1&1&1@1&1&1)] then An = [ 8(3^( 1)&3^( 1)&3^( 1)@3^( 1)&3^( 1)&3^( 1)@3^( 1)&3^( 1)&3^( 1) )] for 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.