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જો શ્રેણિક $A = \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&{ - 2}\\a&2&b\end{array}} \right]$ એે સમીકરણ $AA^T=9I $ નું સમાધાન કરે છે,જયાં $ I$ એ $3×3$ એકમ શ્રેણિક છે,તો ક્રમયુકત જોડ $(a,b)=$
$(-2,-1)$
$(2,-1)$
$(-2,1)$
$(2,1)$
Solution
${{\rm{A}}{{\rm{A}}^ \top } = 9{\rm{I}}}$
$\left[ {\begin{array}{*{20}{c}}
1&2&2\\
2&1&{ – 2}\\
a&2&b
\end{array}} \right]\left[ {\begin{array}{*{20}{r}}
1&2&{\rm{a}}\\
2&1&2\\
2&{ – 2}&b
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
9&0&0\\
0&9&0\\
0&0&9
\end{array}} \right]$
${{\rm{a}} + 4 + 2{\rm{b}} = 0 \Rightarrow {\rm{a}} + 2{\rm{b}} = – 4}$ ……$(i)$
${2{\rm{a}} + 2 – 2{\rm{b}} = 0 \Rightarrow {\rm{a}} + 2{\rm{b}} = – 1}$ …….$(ii)$
${{\rm{ From (i) and (ii) }}}$
${3{\rm{b}} = – 3 \Rightarrow {\rm{b}} = – 1}$
${{\rm{a}} = – 2}$
