7.3-bo'lim : Kengaytirilgan matritsalar
1. Quyidagi kengaytirilgan matritsa uchun ko'rsatilgan elementar qator amallarini bajaring.
\[\left[ {\begin{array}{rrr|r}4&{ - 1}&3&5\\0&2&5&9\\{ - 6}&1&{ - 3}&{10}\end{array}} \right]\]- \(8{R_{\,1}}\)
- \({R_{\,2}} \leftrightarrow {R_{\,3}}\)
- \({R_{\,2}} + 3{R_{\,1}} \to {R_{\,2}}\)
a Yechim
Ushbu operatsiya bizga kengaytirilgan matritsaning 1-qatoridagi barcha yozuvlarni 8 ga ko'paytirishni aytmoqda, shuning uchun buni qilaylik.
\[\left[ {\begin{array}{rrr|r}4&{ - 1}&3&5\\0&2&5&9\\{ - 6}&1&{ - 3}&{10}\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{8{R_{\,1}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rrr|r}{32}&{ - 8}&{24}&{40}\\0&2&5&9\\{ - 6}&1&{ - 3}&{10}\end{array}} \right]\]b Yechim
Ushbu operatsiya bizga kengaytirilgan matritsaning 2 va 3-qatorlarini almashtirishni aytadi. Mana o'sha ish.
\[\left[ {\begin{array}{rrr|r}4&{ - 1}&3&5\\0&2&5&9\\{ - 6}&1&{ - 3}&{10}\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{{R_{\,2}} \leftrightarrow {R_{\,3}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rrr|r}4&{ - 1}&3&5\\{ - 6}&1&{ - 3}&{10}\\0&2&5&9\end{array}} \right]\,\]c Yechim
Ushbu operatsiyani bajarish uchun biz 2-qatorni 2-qatordagi asl yozuvlarni olish natijalari bilan almashtiramiz va ularga 1-qatordagi yozuvlarning 3 barobarini qo'shamiz.
\[\left[ {\begin{array}{rrr|r}4&{ - 1}&3&5\\0&2&5&9\\{ - 6}&1&{ - 3}&{10}\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{{R_{\,2}} + 3{R_{\,1}} \to {R_{\,2}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rrr|r}4&{ - 1}&3&5\\{12}&{ - 1}&{14}&{24}\\{ - 6}&1&{ - 3}&{10}\end{array}} \right]\,\]Quyida ushbu operatsiya uchun individual hisob-kitoblar keltirilgan.
\[\begin{align*}& {\mbox{Column 1 }}:\,\,\,0 + 3\left( 4 \right) = 12\\ & {\mbox{Column 2 }}:\,\,\,2 + 3\left( { - 1} \right) = - 1\\ & {\mbox{Column 3 }}:\,\,\,5 + 3\left( 3 \right) = 14\\ & {\mbox{Column 4 }}:\,\,\,9 + 3\left( 5 \right) = 24\end{align*}\]