Ex 3.2

Chapter 3 Class 12 Matrices
Serial order wise

### Transcript

Ex 3.2, 13 If F (x) = [β 8(cosβ‘π₯&γβsinγβ‘π₯&0@sinβ‘π₯&cosβ‘π₯&0@0&0&1)] , Show that F(x) F(y) = F(x + y) We need to show F(x) F(y) = F(x + y) Solving L.H.S. Given F(x) = [β 8(cosβ‘π₯&γβsinγβ‘π₯&0@sinβ‘π₯&cosβ‘π₯&0@0&0&1)] Finding F(y) Replacing x by y in F(x) F(y) = [β 8(πππβ‘π&γβπππγβ‘π&π@πππβ‘π&πππβ‘π&π@π&π&π)] Now, F(x) F(y) = [β 8(cosβ‘π₯&γβsinγβ‘π₯&0@sinβ‘π₯&cosβ‘π₯&0@0&0&1)] [β 8(cosβ‘π¦&γβsinγβ‘π¦&0@sinβ‘π¦&cosβ‘π¦&0@0&0&1)] = [β 8(cosβ‘π₯ cosβ‘π¦+(γβsinγβ‘π₯ ) sinβ‘γπ¦+0 γ &cosβ‘γπ₯(βsinβ‘γπ¦)+(βsinβ‘γπ₯)γcos yγβ‘γ+ 0γ γ γ γ&0+0+0Γ1@sinβ‘γπ₯ cosβ‘γπ¦+cosβ‘γπ₯ sinβ‘γπ¦+0γ γ γ γ&sinβ‘π₯ (βsinβ‘γπ¦)+γ cosβ‘γπ₯ cosβ‘γπ¦+0γ γ&0+0+0Γ1@0Γcosβ‘γπ¦ +0Γsinβ‘γπ¦+0Γ1γ γ&0Γ(βsinβ‘γπ¦)+0Γcosβ‘γπ¦+0γ γ&0+0+1Γ1)] = [β 8(cosβ‘π₯ cosβ‘π¦ γβsinγβ‘π₯.sinβ‘γπ¦ γ &γβcosγβ‘γπ₯ sinβ‘γπ¦βsinβ‘γπ₯ cosβ‘π¦ γ γ γ&0@sinβ‘γπ₯ cosβ‘γπ¦+cosβ‘γπ₯ sinβ‘π¦ γ γ γ&βsinβ‘π₯ sinβ‘γπ¦+γ cosβ‘γπ₯ cosβ‘π¦ γ&0@0&0&1)] = [β 8(cosβ‘γ(π₯+π¦)γ &γβ[cosγβ‘γπ₯ sinβ‘γπ¦+sinβ‘γπ₯ cosβ‘γπ¦]γ γ γ γ&0@sinβ‘γ(π₯+π¦)γ&cosβ‘π₯ cosβ‘γπ¦ βγ sinβ‘γπ₯ sinβ‘π¦ γ&0@0&0&1)] = [β 8(πππβ‘γ(π+π)γ &βπππβ‘γ(π+π)γ&π@πππβ‘γ(π+π)γ&πππβ‘γ(π+π)γ&π@π&π&π)] We know that cos x cos y β sin x sin y = cos (x + y) & sin x cos y + cos x sin y = sin (x + y) Solving R.H.S F(x + y) Replacing x by (x + y) in F(x) = [β 8(cosβ‘γ(π₯+π¦)γ &βsinβ‘γ(π₯+π¦)γ&0@sinβ‘γ(π₯+π¦)γ&cosβ‘γ(π₯+π¦)γ&0@0&0&1)] = L.H.S. Hence proved