Question Bank • QB_EM_ADV_016
MSQ • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Linear and orthogonal transformations
Question Statement:
Which of the following matrices is/are orthogonal?
Options:
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(A) $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
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(B) $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
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(C) $\frac{1}{3}\begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \end{pmatrix}$
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(D) $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$
Step-by-Step Solution & Explanation
A real square matrix $Q$ is orthogonal if $Q^T Q = I$ (its rows/columns form an orthonormal basis):
1. **Option A**: $R^T R = \begin{pmatrix} \cos^2\theta + \sin^2\theta & 0 \\ 0 & \sin^2\theta + \cos^2\theta \end{pmatrix} = I$ (Orthogonal).
2. **Option B**: $B^T = B$ and $B^2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I$ (Orthogonal).
3. **Option C**: Let $M = \frac{1}{3} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ -2 & 2 & -1 \end{pmatrix}$. Norm of each row is $\frac{\sqrt{1+4+4}}{3} = 1$, and inner products between distinct rows are all $(2+2-4)/9 = 0$ (Orthogonal).
4. **Option D**: First row has norm $\sqrt{1^2 + 1^2} = \sqrt{2} \neq 1$, so it is not orthogonal.
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