Learn All Concepts of Chapter 3 Class 12 Matrices - FREE. Check - Matrices Class 12 - Full video

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.

Examples

Example 1

Example 2

Example 3

Example 4

Example 5 Important

Example 6

Example 7

Example 8

Example 9

Example 10

Example 11

Example 12

Example 13

Example 14

Example 15

Example 16

Example 17

Example 18 Important

Example 19

Example 20 Important

Example 21

Example 22 Important

Example 23 Not in Syllabus - CBSE Exams 2021

Example 24 Important Not in Syllabus - CBSE Exams 2021

Example 25 Important Not in Syllabus - CBSE Exams 2021

Example 26 Important You are here

Example 27 Important

Example 28

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.