Question

The total number of matrices
$A=\left(\begin{array}{ccc}0& 2x& 2x\\ 2y& y& -y\\ 1& -1& 1\end{array}\right),(x,y\in R,x\ne y)$ for which $A^{T}A=3I_3$ is:

• A. $6$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (D)

$4$

$A^{T}A=3I_3$

$\left(\begin{array}{ccc}0& 2x& 2x\\ 2y& y& -y\\ 1& -1& 1\end{array}\right)\left(\begin{array}{ccc}0& 2y& 1\\ 2x& y& -1\\ 2x& -y& -y\end{array}\right)=\left(\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right)$

$\left(\begin{array}{ccc}8x^2& 0& 0\\ 0& 6y^2& 0\\ 0& 0& 3\end{array}\right)=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]$

$8x^2=3;6y^2=3\Rightarrow x^2=3/8;y^2=1/2\Rightarrow x=\pm \sqrt{\frac{3}{8}};y=\pm \sqrt{\frac{1}{2}}$

Thus total number of matrices can be formed with these values are $4$