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  • Help me please, - Let - Matrices and Determinants - JEE Main

 

Let A=\begin{pmatrix} 0 & 2q & r\\ p & q & -r\\ p& -q & r \end{pmatrix}.\: If\: \: AA^{T}=I_{3},\: Then \: \left | p \right |\: is\: :

  • Option 1)

     

    \frac{1}{\sqrt2}

  • Option 2)

     

    \frac{1}{\sqrt6}

  • Option 3)

     

    \frac{1}{\sqrt3}

  • Option 4)

     

    \frac{1}{\sqrt5}

Answers (1)

best_answer

 

Transpose of a Matrix -

The matrix obtained from any given matrix A, by interchanging its rows and columns.

- wherein

 

 

Multiplication of matrices -

 

\huge \large\bigl(\begin{smallmatrix} a_{11} &a_{12} &a_{13} \\ a_{21}& a_{22} &a_{23} \\ a_{31}&a_{32} &a_{33} \end{smallmatrix}\bigr)\times \bigl(\begin{smallmatrix} b_{11} &b_{12} &b_{13} \\ b_{21}& b_{22} &b_{23} \\ b_{31}&b_{32} &b_{33} \end{smallmatrix}\bigr)=

\huge \large\bigl(\begin{smallmatrix} a_{11}b_{11}+a_{12}b_{21}+a_{13}b_{31} \:\:\:&a_{11}b_{12}+a_{12}b_{22}+a_{13}b_{32} &\:\:\:a_{11}b_{13}+a_{12}b_{23}+a_{13}b_{33} \\ a_{21}b_{11}+a_{22}b_{21}+a_{23}b_{31}& \:a_{21}b_{12}+a_{22}b_{22}+a_{23}b_{32} &\:\:a_{21}b_{13}+a_{22}b_{23}+a_{23}b_{33} \\ a_{31}b_{11}+a_{32}b_{21}+a_{33}b_{31}&a_{31}b_{12}+a_{32}b_{22}+a_{33}b_{32} &\:\:a_{31}b_{13}+a_{32}b_{23}+a_{33}b_{33} \end{smallmatrix}\bigr)

-

 

AA^{T}=I

\Rightarrow \begin{bmatrix} 0 &2q &r \\ p& q &-r \\ p& -q& r \end{bmatrix}\begin{bmatrix} 0 &p &p \\ 2q&q &-q \\ r&-r &r \end{bmatrix}=\begin{bmatrix} 1 &0 &0 \\ 0&1 &0 \\ 0&0 &1 \end{bmatrix}

\begin{bmatrix} 4q^{2}+r^{2} &2q^{2} -r^{2} &-2q^{2} +r^{2}\\ 2q^{2}-r^{2}& p^{2}+q^{2}+r^{2} & p^{2}-q^{2}-r^{2}\\ -2q^{2}+r^{2}& p^{2}-q^{2} -r^{2}& p^{2}+q^{2}+r^{2} \end{bmatrix}=\begin{bmatrix} 1 &0 & 0\\ 0& 1 &0 \\ 0&0 &1 \end{bmatrix}

\Rightarrow p^{2}+q^{2}+r^{2}=4q^{2}+r^{2}=1  and ,

2q^{2}-r^{2}=0\: ,\: p^{2}-q^{2}-r^{2}=0

Hence, \left | p \right |=\frac{1}{\sqrt{2}}


Option 1)

 

\frac{1}{\sqrt2}

Option 2)

 

\frac{1}{\sqrt6}

Option 3)

 

\frac{1}{\sqrt3}

Option 4)

 

\frac{1}{\sqrt5}

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