Q. Which of the following is an orthogonal matrix?
A) $\begin{bmatrix}
\frac{6}{7} & \frac{2}{7} & \frac{- 3}{7} \\
\frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\
\frac{3}{7} & \frac{- 6}{7} & \frac{2}{7}
\end{bmatrix}$
B) $\begin{bmatrix}
\frac{6}{7} & \frac{2}{7} & \frac{3}{7} \\
\frac{2}{7} & \frac{- 3}{7} & \frac{6}{7} \\
\frac{3}{7} & \frac{6}{7} & \frac{- 2}{7}
\end{bmatrix}$
C) $\begin{bmatrix}
- \frac{6}{7} & - \frac{2}{7} & \frac{- 3}{7} \\
\frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\
\frac{- 3}{7} & \frac{6}{7} & \frac{2}{7}
\end{bmatrix}$
D) $\begin{bmatrix}
\frac{6}{7} & \frac{- 2}{7} & \frac{3}{7} \\
\frac{2}{7} & \frac{2}{7} & \frac{- 3}{7} \\
\frac{- 6}{7} & \frac{2}{7} & \frac{3}{7}
\end{bmatrix}$
Explanation
(a)
Matrix $\begin{bmatrix}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{bmatrix}$ is orthogonal if
$\sum_{}^{}{a_{i}^{2} =}\sum_{}^{}{b_{i}^{2} =}\sum_{}^{}{c_{i}^{2} = 1;\ }\sum_{}^{}{a_{i}b_{i} =}\sum_{}^{}{b_{i}c_{i} =}\sum_{}^{}{c_{i}a_{i} =}0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.