If X=≤ft[{array}{ll}1 & 0 \\ 0 & 1{array}] and Y=≤ft[{array}{cc}0 & 1 \\ -1 & 0{array}] and B=≤ft[{array}{cc} θ & θ \\ - θ & θ{array}] then B equals :

Correct Answer is : Xcos⁡θ+Ysin⁡θX θ+Y θXcosθ+Ysinθ

Correct Answer is : Xcos⁡θ+Ysin⁡θX θ+Y θXcosθ+Ysinθ

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COMEDK 2025 Evening Solved Paper

Question : 127 of 180

Marks: +1, -0

X=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

Y=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]

B=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]

B

equals :

Solution:

Concept:

Express matrix

B

as a linear combination of

X

and

Y

by comparing entries.

Explanation:

Compute

\cos\theta \, X + \sin\theta \, Y

\cos\theta \begin{pmatrix}1&0\\0&1\end{pmatrix} + \sin\theta \begin{pmatrix}0&1\\-1&0\end{pmatrix}

= \begin{pmatrix}\cos\theta & 0\\0 & \cos\theta\end{pmatrix} + \begin{pmatrix}0 & \sin\theta\\-\sin\theta & 0\end{pmatrix}

= \begin{pmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{pmatrix}

, which is exactly

B

Thus

B = X\cos\theta + Y\sin\theta

Answer:

Option D:

X \cos\theta + Y \sin\theta

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