Question

If \( A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \) then \( A^2 - (2 \cos \alpha) A \) is equal to: (Where I is identity matrix of order 2)

• A. A
• B. -A
• C. 2A + I
• D. -I

Hint:

Every \( 2 \times 2 \) matrix satisfies its own characteristic equation \( A^2 - \text{tr}(A)A + \det(A)I = 0 \), which is a very powerful tool for such problems.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We use the characteristic equation of a \( 2 \times 2 \) matrix, given by \( A^2 - \text{tr}(A)A + \det(A)I = 0 \).

Step 2: Mathematical Calculation:
For \( A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \):
Trace \( \text{tr}(A) = \cos \alpha + \cos \alpha = 2 \cos \alpha \).
Determinant \( \det(A) = (\cos \alpha)(\cos \alpha) - (\sin \alpha)(-\sin \alpha) = \cos^2 \alpha + \sin^2 \alpha = 1 \).
Substituting into the characteristic equation:
\( A^2 - (2 \cos \alpha)A + 1 \cdot I = 0 \)
\( A^2 - (2 \cos \alpha)A = -I \).

Step 3: Final Answer:
Thus, the expression is equal to \( -I \).