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Question: Let E(a) = \(\begin{bmatrix} \cos^{2}\alpha & \cos\alpha\sin\alpha \\ \cos\alpha\sin\alpha & \sin^{2...

Let E(a) = [cos⁡2αcos⁡αsin⁡αcos⁡αsin⁡αsin⁡2α]\begin{bmatrix} \cos^{2}\alpha & \cos\alpha\sin\alpha \\ \cos\alpha\sin\alpha & \sin^{2}\alpha \end{bmatrix}. If a and b differs by an odd multiple of p/2, then E(a) E(b) is a –

A

Null matrix

B

Unit matrix

C

Diagonal matrix

D

Orthogonal matrix

Answer

Null matrix

Explanation

Solution

We have

E(a) E(b) = [cos⁡2αcos⁡αsin⁡αcos⁡αsin⁡αsin⁡2α]\begin{bmatrix} \cos^{2}\alpha & \cos\alpha\sin\alpha \\ \cos\alpha\sin\alpha & \sin^{2}\alpha \end{bmatrix}

= [cos⁡αcos⁡βcos⁡(α−β)cos⁡αsin⁡βcos⁡(α−β)sin⁡αcos⁡βcos⁡(α−β)sin⁡αsin⁡βcos⁡(α−β)]\begin{bmatrix} \cos\alpha\cos\beta\cos(\alpha - \beta) & \cos\alpha\sin\beta\cos(\alpha - \beta) \\ \sin\alpha\cos\beta\cos(\alpha - \beta) & \sin\alpha\sin\beta\cos(\alpha - \beta) \end{bmatrix}As a and b differ by an odd multiple of p/2,

a – b = (2n + 1) p/2 for some As a and b integer n. Thus,

cos [(2n + 1)p/2] = 0

\ E (a) E (b) = O.