# Example 26 - Chapter 3 Class 12 Matrices

Last updated at Jan. 17, 2020 by Teachoo

Last updated at Jan. 17, 2020 by Teachoo

Transcript

Example 26 If A = [■8( cos" θ" &sin" θ" @−sin "θ" &cos" θ" )] , then prove that An = [■8(cos n"θ" &sin n"θ" @−sin "nθ" &cos n"θ" )],n ∈ N. We shall prove the result by using mathematical induction. Step 1: Let P(n) : If A = [■8(cos "θ" &sin" θ" @−sin "θ" &cos" θ" )] then An = [■8(cos n"θ" &sin n"θ" @−sin" nθ" &cos n"θ" )] ,n ∈ N. Step 2: Prove for n = 1 For n = 1 L.H.S = A1 = A = [■8(cos" θ" &sin" θ" @−sin" θ" &cos" θ" )], R.H.S = [■8(cos" 1θ" &sin "1θ" @−sin 1"θ" &cos "1θ" )] =[■8(cos" θ" &sin" θ" @−sin" θ" &cos" θ" )] 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 Assume that P (k) is true P(k) : If A = [■8(cos "θ" &sin "θ" @−sin "θ" &cos "θ" )], then Ak = [■8(cos "kθ" &sin k"θ" @−sin k"θ" &cos k"θ" )] , where k ∈ N We will have to prove that P( k +1) is true P(k + 1) : If A = [■8(cos" θ" &sin" θ" @−sin" θ" &cos" θ" )] , then we need to prove Ak+1 = [■8(cos" (k + 1)θ" &sin" (k + 1)θ" @−sin" (k + 1)θ" &cos" (k + 1)θ" )] Taking L.H.S Ak+1 = Ak . A = [■8(cos "kθ" &sin k"θ" @−sin k"θ" &cos k"θ" )] [■8(cos "θ" &sin "θ" @−sin "θ" &cos" θ" )] Ak+1 = [■8(cos" (k + 1)θ" &sin" (k + 1)θ" @−sin" (k + 1)θ" &cos" (k + 1)θ" )] Taking L.H.S Ak+1 = Ak . A = [■8(cos "kθ" &sin k"θ" @−sin k"θ" &cos k"θ" )] [■8(cos "θ" &sin "θ" @−sin "θ" &cos" θ" )] =[■8(cos" kθ" (cos" θ" )+sin k"θ" (−sin" θ)" &cos kθ(sin "θ)" +sin kθ(cos" θ" )@−sin" kθ" (cos "θ" )+cos k"θ" (−sin "θ)" &−sin "kθ(" sin "θ" )+ cos k"θ" (cos "θ" ))] = [■8(cos" kθ" cos" θ" −sin k"θ" sin" θ" &cos kθ sin "θ" +sin kθ cos" θ" @−sin" kθ" cos "θ" −cos k"θ" sin "θ" &−sin "kθ" sin "θ" + cos k"θ" cos "θ" )] = [■8(cos" kθ" cos" θ" −sin k"θ" sin" θ" &sin "θ" cos kθ +sin kθ cos" θ" @−(sin" kθ" cos "θ" +cos k"θ" sin "θ)" &cos k"θ" cos "θ" − sin "kθ" sin "θ" )] Using cos (x + y) = cos x cos y – sin x sin y Sin (x + y) = sin x cos y + cos y sin x = [■8(cos"(" k"θ + θ)" &sin "(" k"θ + θ)" @−sin" (" k"θ + θ)" &"cos (" k"θ + θ)" )] = [■8(cos"(" k" + 1)θ" &sin" (" k" + 1)θ" @−sin "(" k" + 1)θ" &"cos (" k" + 1)θ" )] = R.H.S Thus P (k + 1) is true ∴ By the principal of mathematical induction , P(n) is true for n ∈ N Thus, if A = [■8(cos "θ" &sin" θ" @−sin "θ" &cos" θ" )] then An = [■8(cos n"θ" &sin n"θ" @−sin" nθ" &cos n"θ" )] for all n ∈ N.

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Example 23 Deleted for CBSE Board 2021 Exams only

Example 24 Important Deleted for CBSE Board 2021 Exams only

Example 25 Important Deleted for CBSE Board 2021 Exams only

Example 26 Important You are here

Example 27 Important

Example 28

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.