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EEJ MAIN Mathematics QUESTION #6944
Question 1

Count the total number of $3\times3$ matrices of the form $A=\begin{pmatrix}0&2y&1\\2x&y&-1\\2x&-y&1\end{pmatrix}$ with $x,y\in\mathbb{R},\ x\neq y$, satisfying $A^TA=3I_3$.

  • 2
  • 3
  • 6
  • 4✔️
Correct Answer Explanation

Compute $A^TA = 3I_3$ by equating columns to be orthogonal with magnitude $\sqrt{3}$.

Column 1 norm$^2$: $0+(2x)^2+(2x)^2=8x^2=3 \Rightarrow x^2=\frac{3}{8}$: 2 values of $x$.

Column 2 norm$^2$: $4y^2+y^2+y^2=6y^2=3 \Rightarrow y^2=\frac{1}{2}$: 2 values of $y$.

With $x\neq y$: we need to discard cases where $x=y$. Since $x^2=3/8$ and $y^2=1/2$, $x\neq\pm y$ (different magnitudes), so all $2\times2=4$ combinations are valid.

Total $= \mathbf{4}$