Question

If the matrix \[ \begin{pmatrix} 0 & a & a
2b & b & -b
c & -c & c \end{pmatrix} \] is orthogonal, then the values of \( a,b,c \) are:

• A. \( a=\pm\frac{1}{\sqrt{3}}, \, b=\pm\frac{1}{\sqrt{6}}, \, c=\pm\frac{1}{\sqrt{2}} \)
• B. \( a=\pm\frac{1}{\sqrt{2}}, \, b=\pm\frac{1}{\sqrt{6}}, \, c=\pm\frac{1}{\sqrt{3}} \)
• C. \( a=\pm\frac{1}{\sqrt{2}}, \, b=\pm\frac{1}{\sqrt{6}}, \, c=\pm\frac{1}{\sqrt{3}} \)
• D. \( a=\pm\frac{1}{\sqrt{3}}, \, b=\pm\frac{1}{\sqrt{6}}, \, c=\pm\frac{1}{\sqrt{3}} \)

Hint:

For orthogonal matrices: \begin{itemize} \item Normalize rows using sum of squares. \item Check perpendicularity via dot products. \end{itemize}

Answer 1

The Correct Option is A