Question

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

• A. 6
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (D)

$A^{T}A =3I_{3} $
$ \begin{pmatrix}0&2x&2x\\ 2y&y&-y\\ 1&-1&1\end{pmatrix}\begin{pmatrix}0&2y&1\\ 2x&y&-1\\ 2x&-y&1\end{pmatrix} = \begin{pmatrix}3&0&0\\ 0&3&0\\ 0&0&3\end{pmatrix} $
$ \begin{pmatrix}8x^{2}&0&0\\ 0&6y^{2}&0\\ 0&0&3\end{pmatrix} = \begin{bmatrix}3&0&0\\ 0&3&0\\ 0&0&3\end{bmatrix} $
$ 8x^{2}= 3 $
$ 6y^{2}=3 $
$ x^{2} =3/8 $
$ y^{2} =1/2 $
$ x = \pm \sqrt{\frac{3}{8}} ; y= \pm \sqrt{\frac{1}{2}} $