# Example 26 - Chapter 3 Class 12 Matrices

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 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" )] =[■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" )] = [■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 Hence, 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

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 Important

Example 20

Example 21

Example 22 Important

Example 23

Example 24

Example 25

Example 26 You are here

Example 27 Important

Example 28 Important

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.