Question

2. Compute the following:a b a b-1 4-6]「12 7 6(iüi)8 5 168 0 5 (iv)2 8 5 3 2 4Lsin2 xcos2 x」Lcos2 xsin2x

Answer 1

(i)

Adding the matrices, we get:

[ a b −b a ]+[ a b b a ]=[ 2a 2b 0 2a ]

Thus, the required matrix is [ 2a 2b 0 2a ] .

(ii)

Adding the matrices,

[ a 2 + b 2 b 2 + c 2 a 2 + c 2 a 2 + b 2 ]+[ 2ab 2bc −2ac −2ab ]=[ a 2 + b 2 +2ab b 2 + c 2 +2bc a 2 + c 2 −2ac a 2 + b 2 −2ab ] =[ ( a+b ) 2 ( b+c ) 2 ( a−c ) 2 ( a−b ) 2 ]

Thus, the required result is [ ( a+b ) 2 ( b+c ) 2 ( a−c ) 2 ( a−b ) 2 ] .

(iii)

Adding the matrices,

[ −1 4 −6 8 5 16 2 8 5 ]+[ 12 7 6 8 0 5 3 2 4 ]=[ −1+12 4+7 −6+6 8+8 5+0 16+5 2+3 8+2 5+4 ] =[ 11 11 0 16 5 21 5 10 9 ]

Thus, the result is given as [ 11 11 0 16 5 21 5 10 9 ] .

(iv)

Adding the matrices,

[ cos 2 x sin 2 x sin 2 x cos 2 x ]+[ sin 2 x cos 2 x cos 2 x sin 2 x ]=[ cos 2 x+ sin 2 x sin 2 x+ cos 2 x sin 2 x+ cos 2 x cos 2 x+ sin 2 x ]

Using the identity sin 2 x+ cos 2 x=1 , we get:

[ cos 2 x sin 2 x sin 2 x cos 2 x ]+[ sin 2 x cos 2 x cos 2 x sin 2 x ]=[ cos 2 x+ sin 2 x sin 2 x+ cos 2 x sin 2 x+ cos 2 x cos 2 x+ sin 2 x ] =[ 1 1 1 1 ]

Thus, the required result is [ 1 1 1 1 ] .