Q.2

If 𝑃 = ( − 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼) and 𝑃 + 𝑃 𝑇 = I, the value of α (0 ≤ 𝛼 ≤ 𝜋⁄2) is

(A) 𝜋/2

(B) 𝜋/3

(C) 3𝜋/2

(D) 0

Introduction

In competitive exams like IIT JAM, BSc, BCA, and MSc entrance tests, matrix questions involving transpose and identity matrices are very common.

In this article, we solve a standard and important problem based on orthogonal matrices and trigonometric identities.

Question

If

P = \begin{pmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{pmatrix}

and P + P^T = I

then find the value of α, where 0 ≤ α ≤ 2π

Options

• (A) π/2
• (B) π/3
• (C) 3π/2
• (D) 0

Step-by-Step Solution

Step 1: Find Transpose of P

Transpose means interchange rows and columns.

P^T = \begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{pmatrix}

Step 2: Add P and Pᵀ

P + P^T = \begin{pmatrix} \cos \alpha + \cos \alpha & \sin \alpha – \sin \alpha \\ -\sin \alpha + \sin \alpha & \cos \alpha + \cos \alpha \end{pmatrix}
= \begin{pmatrix} 2\cos \alpha & 0 \\ 0 & 2\cos \alpha \end{pmatrix}

Step 3: Compare with Identity Matrix

Identity matrix:

I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

So, 2\cos α = 1 ⇒ \cos α = 1/2

Step 4: Find α

\cos α = 1/2 ⇒ α = π/3

This value lies in the given range 0 ≤ α ≤ 2π.

Correct Answer

✅ Option (B) π/3

Explanation of All Options

Option (A): π/2

❌ \cos(π/2) = 0 ⇒ P + P^T = 0 ≠ I

Option (B): π/3

✅ \cos(π/3) = 1/2 ⇒ P + P^T = I

Option (C): 3π/2

❌ Not in the given range. Also, \cos(3π/2) = 0

Option (D): 0

❌ \cos(0) = 1 ⇒ P + P^T = 2I ≠ I

Final Conclusion

The matrix simplifies beautifully due to symmetry. Only Option (B) satisfies both the equation and the given range.