Question

If A=[cos⁡2θ−sin⁡2θsin⁡2θcos⁡2θ] and A+AT=I Where I is the unit matrix of 2×2 and AT is the transpose of A, then the value of θ is equal to:

• A. (A) 3π2
• B. (B) π
• C. (C) π3
• D. (D) π6

Answer 1

Answer: (D)

(D) π6

Answer 2

Explanation:
Given,A=[cos⁡2θ−sin⁡2θsin⁡2θcos⁡2θ] A+AT=1The transpose of matrix A is given by, AT=[cos⁡2θsin⁡2θ−sin⁡2θcos⁡2θ]As, A+AT=I [cos⁡2θ−sin⁡2θsin⁡2θcos⁡2θ]+[cos⁡2θsin⁡2θ−sin⁡2θcos⁡2θ]=I ⇒[2cos⁡2θ002cos⁡2θ]=[1001] As we know that,If two matrices A and B are equal then their corresponding elements are also equal.∴2cos⁡2θ=1⇒cos⁡2θ=12⇒cos⁡2θ=cos⁡60∘⇒2θ=π3[∵cos⁡60∘=π3]∴θ=π6So, the value of θ is π6.Hence, the correct option is (D).