The total number of matrices 

A=\begin{pmatrix} 0 &2y &1 \\ 2x&y & -1\\ 2x & -y& 1 \end{pmatrix}  ,\left ( x,y\epsilon R,\:x\neq y \right ) for which A^{T}A=3I_{3}  is :

  • Option 1)

    2

  • Option 2)

    3

  • Option 3)

    6

  • Option 4)

    4

 

Answers (1)

A=\begin{pmatrix} 0 &2y &1 \\ 2x&y & -1\\ 2x & -y& 1 \end{pmatrix}  ,\left ( x,y\epsilon R,\:x\neq y \right )

 

A^{T}A=3I_{3}

 

AA^{T}=\begin{pmatrix} 0 &2y &1 \\ 2x&y & -1\\ 2x & -y& 1 \end{pmatrix}\begin{pmatrix} 0 &2x &2x \\ 2y& y &-y \\ 1& -1& 1 \end{pmatrix}

\begin{pmatrix} 4y^{2}+1 &2y^{2}-1 &-2y^{2}+1 \\ 2y^{2}-1 & 4x^{2}+y^{2}+1 & 4x^{2}-y^{2}-1\\ -2y^{2}+1&4x^{2}-y^{2}-1 &4x^{2}+y^{2}+1 \end{pmatrix}=\begin{pmatrix} 3 &0 &0 \\ 0& 3 & 0\\ 0 &0 &3 \end{pmatrix}

\\4y^{2}+1=3,\therefore y^{2}=1/2\\\\\:4x^{2}+y^{2}+1=3

4x^{2}+y^{2}=2

\\4x^{2}=2-1/2\\\\\:4x^{2}=3/2

x^{2}=3/8

x,y=\left ( \pm\frac{1}{\sqrt{2}},\pm \sqrt{3/8} \right )

4 matrices 

 


Option 1)

2

Option 2)

3

Option 3)

6

Option 4)

4

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