The total number of matrices \(A = \begin{pmatrix} 0 & 2y & 1 \\ 2x & y & -1 \\ 2x & -y & 1 \end{pmatrix}\) \((x, y \in R,\ x \ne y)\) for which \(A^{T}A = 3I_3\) is :-
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- 3
- 4
- 2
- 6
The Correct Option is B
Solution and Explanation
\(A^TA = 3I\) needs \(|c_i|^2 = 3\) and \(c_i\cdot c_j = 0\).
\(|c_1|^2 = 8x^2 = 3 \Rightarrow x^2 = \frac38\); \(|c_2|^2 = 6y^2 = 3 \Rightarrow y^2 = \frac12\); \(|c_3|^2 = 3\) holds.
Dot products: \(c_1\cdot c_2 = 2xy - 2xy = 0\), \(c_1\cdot c_3 = -2x+2x = 0\), \(c_2\cdot c_3 = 2y - y - y = 0\), all hold automatically.
So \(x=\pm\sqrt{3/8}\), \(y=\pm\sqrt{1/2}\): 4 combinations, and since \(x^2\ne y^2\), \(x\ne y\) always. Number of matrices = 4.
Answer: (2) 4
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