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Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMatrices3 Marks
Question
Find the values of $\mathrm{x}, \mathrm{y}, \mathrm{z}$ if the matrix $A=\left[\begin{array}{ccc}0 & 2 y & z \\ x & y & -z \\ x & -y & z\end{array}\right]$ satisfy the equation $\mathrm{A}^{\prime} \mathrm{A}=\mathrm{I}$.
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Answer

It is given that: $A = \left[ {\begin{array}{*{20}{c}} 0&{2y}&z \\ x&y&{ - z} \\ x&{ - y}&z \end{array}} \right]$
$ \Rightarrow A' = \left[ {\begin{array}{*{20}{c}} 0&x&x \\ {2y}&y&{ - y} \\ z&{ - z}&z \end{array}} \right]$
From the given question, $A'.A = I \Rightarrow \left[ {\begin{array}{*{20}{c}} 0&x&x \\ {2y}&y&{ - y} \\ z&{ - z}&z \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&x&x \\ {2y}&y&{ - y} \\ z&{ - z}&z \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right]$
$ \Rightarrow \left[ {\begin{array}{*{20}{c}} {0 + {x^2} + {x^2}}&{0 + xy - xy}&{0 - xy + xz} \\ {0 + xy - xy}&{4{y^2} + {y^2} + {y^2}}&{2yz - yz - yz} \\ {0 - zx + zx}&{2yz - yz - yz}&{{z^2} + {z^2} + {z^2}} \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right]$
$ \Rightarrow \left[ {\begin{array}{*{20}{c}} {2{x^2}}&0&0 \\ 0&{6{y^2}}&0 \\ 0&0&{3{z^2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right]$
Equating corresponding entries,we obtain
$2x^2 = 1 \Rightarrow {x^2} = \frac{1}{2} \Rightarrow x = \pm \frac{1}{{\sqrt 2 }}$
And $6y^2 = 1 \Rightarrow {y^2} = \frac{1}{6} \Rightarrow y = \pm \frac{1}{{\sqrt 6 }}$
And $3z^2 = 1 \Rightarrow {z^2} = \frac{1}{3} \Rightarrow z = \pm \frac{1}{{\sqrt 3 }}$
$\therefore x = \pm \frac{1}{{\sqrt 2 }},\;y = \pm \frac{1}{{\sqrt 6 }},\;Z = \pm \frac{1}{{\sqrt 3 }}$

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