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



Last updated at Jan. 17, 2020 by Teachoo
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Ex 3.2, 13 If F (x) = [■8(cos𝑥&〖−sin〗𝑥&0@sin𝑥&cos𝑥&0@0&0&1)] , Show that F(x) F(y) = F(x + y) We need to show F(x) F(y) = F(x + y) Taking L.H.S. Given F(x) = [■8(cos𝑥&〖−sin〗𝑥&0@sin𝑥&cos𝑥&0@0&0&1)] Finding F(y) Replacing x by y in F(x) F(y) = [■8(cos𝑦&〖−sin〗𝑦&0@sin𝑦&cos𝑦&0@0&0&1)] Now, F(x) F(y) = [■8(cos𝑥&〖−sin〗𝑥&0@sin𝑥&cos𝑥&0@0&0&1)] [■8(cos𝑦&〖−sin〗𝑦&0@sin𝑦&cos𝑦&0@0&0&1)] = [■8(cos𝑥 cos𝑦+(〖−sin〗𝑥 ) sin〖𝑦+0 〗 &cos〖𝑥(−sin〖𝑦)+(−sin〖𝑥)〖cos y〗〖+ 0〗 〗 〗 〗&0+0+0×1@sin〖𝑥 cos〖𝑦+cos〖𝑥 sin〖𝑦+0〗 〗 〗 〗&sin𝑥 (−sin〖𝑦)+〗 cos〖𝑥 cos〖𝑦+0〗 〗&0+0+0×1@0×cos〖𝑦 +0×sin〖𝑦+0×1〗 〗&0×(−sin〖𝑦)+0×cos〖𝑦+0〗 〗&0+0+1×1)] = [■8(cos𝑥 cos𝑦 〖−sin〗𝑥.sin〖𝑦 〗 &〖−cos〗〖𝑥 sin〖𝑦−sin〖𝑥 cos𝑦 〗 〗 〗&0@sin〖𝑥 cos〖𝑦+cos〖𝑥 sin𝑦 〗 〗 〗&−sin𝑥 sin〖𝑦+〗 cos〖𝑥 cos𝑦 〗&0@0&0&1)] We know that cos x cos y – sin x sin y = cos (x + y) & sin x cos y + cos x sin y = sin (x + y) = [■8(cos〖(𝑥+𝑦)〗 &〖−[cos〗〖𝑥 sin〖𝑦+sin〖𝑥 cos〖𝑦]〗 〗 〗 〗&0@sin〖(𝑥+𝑦)〗&cos𝑥 cos〖𝑦 −〗 sin〖𝑥 sin𝑦 〗&0@0&0&1)] = [■8(cos〖(𝑥+𝑦)〗 &−sin〖(𝑥+𝑦)〗&0@sin〖(𝑥+𝑦)〗&cos〖(𝑥+𝑦)〗&0@0&0&1)] Taking R.H.S F(x + y) Replacing x by (x + y) in F(x) = [■8(cos〖(𝑥+𝑦)〗 &−sin〖(𝑥+𝑦)〗&0@sin〖(𝑥+𝑦)〗&cos〖(𝑥+𝑦)〗&0@0&0&1)] = L.H.S. Hence proved
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