Question

Let E (a) =$\begin{bmatrix} \cos^{2}\alpha & \cos\alpha\sin\alpha \\ \cos\alpha\sin\alpha & \sin^{2}\alpha \end{bmatrix}$. 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 1

Answer: (A)

Null matrix

Answer 2

We have

E(a)(b)=$\left[ \begin{array} { c c } \cos ^ { 2 } \alpha & \cos \alpha \sin \alpha \\ \cos \alpha \sin \alpha & \sin ^ { 2 } \alpha \end{array} \right]$ $\begin{bmatrix} \cos^{2}\beta & \cos\beta\sin\beta \\ \cos\beta\sin\beta & \sin^{2}\beta \end{bmatrix}$

= $\begin{bmatrix} \cos\alpha\cos\beta\cos(\alpha - \beta) & \cos\alpha\sin\beta\cos(\alpha - \beta) \\ \sin\alpha\cos\beta\cos(\alpha - \beta) & \sin\alpha\sin\beta\cos(\alpha - \beta) \end{bmatrix}$

As a and b differ by an odd multiple of p/2, a –b =

(2n 1) p/2 for some integer n. Thus, cos [(2n +1) p/2] = 0

\ E(a) E(b) = 0