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સમીકરણ $\left[ {\begin{array}{*{20}{c}}1&0&1\\{ - 1}&1&0\\0&{ - 1}&1\end{array}} \right]\,\left[ \begin{array}{l}x\\y\\z\end{array} \right] = \left[ \begin{array}{l}1\\1\\2\end{array} \right]$ નો ઉકેલ $(x,y,z)$ = . . .
$(1,\,\,1\,,\,1)$
$(0,\, - 1\,\,2)$
$( - 1,\,\,2,\,\,2)$
$(-1, 0, 2)$
Solution
(d) Let $A = \left[ {\begin{array}{*{20}{c}}1&0&1\\{ – 1}&1&0\\0&{ – 1}&1\end{array}} \right]\, \Rightarrow \,{A^{ – 1}}\left[ {\begin{array}{*{20}{c}}{\frac{1}{2}}&{\frac{{ – 1}}{2}}&{ – 1}\\{\frac{1}{2}}&{\frac{1}{2}}&{\frac{{ – 1}}{2}}\\{\frac{1}{2}}&{\frac{1}{2}}&{\frac{1}{2}}\end{array}} \right]$
$\therefore $ $AX = B \Rightarrow X = {A^{ – 1}}B\,\left[ {\begin{array}{*{20}{c}}{\frac{1}{2}}&{\frac{{ – 1}}{2}}&{ – 1}\\{\frac{1}{2}}&{\frac{1}{2}}&{\frac{{ – 1}}{2}}\\{\frac{1}{2}}&{\frac{1}{2}}&{\frac{1}{2}}\end{array}} \right]\,\,\left[ \begin{array}{l}1\\1\\2\end{array} \right] = \left[ \begin{array}{l} – 1\\{\rm{ }}0\\{\rm{ }}2\end{array} \right]$.
Aliter : $\left[ {\begin{array}{*{20}{c}}1&0&1\\{ – 1}&1&0\\0&{ – 1}&1\end{array}} \right]\,\,\,\,\left[ \begin{array}{l}x\\y\\z\end{array} \right] = \left[ \begin{array}{l}1\\1\\2\end{array} \right]$
$ \Rightarrow $ $\left[ \begin{array}{l}x + 0y + z\\ – x + y + 0z\\0x – y + z\end{array} \right] = \left[ \begin{array}{l}1\\1\\2\end{array} \right]$ ==> $\begin{array}{l}x + z = 1\\ – x + y = 1\\z – y = 2\end{array}$
$ \Rightarrow $ $(x,\,y,\,z) = ( – 1,\,0,\,2)$.
