Question

If \( A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), \( B = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \) and \( C = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \) then

• A. \( C = A\cos\theta - B\sin\theta \)
• B. \( C = A\sin\theta + B\cos\theta \)
• C. \( C = A\sin\theta - B\cos\theta \)
• D. \( C = A\cos\theta + B\sin\theta \)

Hint:

Matrix Operations. Scalar multiplication: Multiply each element by the scalar. Matrix addition: Add corresponding elements. Test the given options by performing the operations.

Answer 1

The Correct Option is D

We are given matrices A, B, and C
Let's test the relationship given in option (D)
$$ A\cos\theta = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \cos\theta = \begin{bmatrix} \cos\theta & 0 \\ 0 & \cos\theta \end{bmatrix} $$ $$ B\sin\theta = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \sin\theta = \begin{bmatrix} 0 & \sin\theta \\ -\sin\theta & 0 \end{bmatrix} $$ Now add these two matrices: $$ A\cos\theta + B\sin\theta = \begin{bmatrix} \cos\theta & 0 \\ 0 & \cos\theta \end{bmatrix} + \begin{bmatrix} 0 & \sin\theta \\ -\sin\theta & 0 \end{bmatrix} $$ $$ = \begin{bmatrix} \cos\theta + 0 & 0 + \sin\theta \\ 0 - \sin\theta & \cos\theta + 0 \end{bmatrix} = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} $$ This resulting matrix is exactly matrix C
Therefore, \( C = A\cos\theta + B\sin\theta \) is the correct relationship
Note that A is the identity matrix I, and C represents a rotation matrix