Consider the matrix f x = cos x - sin x 0 sin x cos x 0 0 0 1 . Given below are two statements : Statement I: f ( - x ) is the inverse of the matrix f ( x ) . Statement II: f ( x ) f ( y ) = f ( x + y ) . In the light of the above statements, choose the correct answer from the options given below

Step-by-step solution

Given: f x = cos x - sin x 0 sin x cos x 0 0 0 1 ⇒ f - x = cos x sin x 0 - sin x cos x 0 0 0 1 ⇒ f x × f - x = cos x - sin x 0 sin x cos x 0 0 0 1 cos x sin x 0 - sin x cos x 0 0 0 1 ⇒ f x × f - x = 1 0 0 0 1 0 0 0 1 ⇒ f x × f - x = I ⇒ f - x = f - 1 x Hence statement-I is correct Now, checking statement II ⇒ f y = cos y - sin y 0 sin y cos y 0 0 0 1 ⇒ f x f y = cos x - sin x 0 sin x cos x 0 0 0 1 cos y - sin y 0 sin y cos y 0 0 0 1 ⇒ f x f y = cos x cos y - sin x sin y - cos x sin y - sin x cos y 0 sin x cos y + c