7.4-bo'lim : Kengaytirilgan matritsa haqida ko'proq
1. Quyidagi tenglamalar sistemasi uchun tizimni kengaytirilgan matritsaga aylantiring va tizimning yechimini aniqlash yoki tizimning nomuvofiq yoki bog'liqligini aniqlash uchun to'ldirilgan matritsa usullaridan foydalaning.
\[\begin{align*}x - 7y & = - 11\\ 5x + 2y & = - 18\end{align*}\]Yechim
Birinchi qadam tenglamalar sistemasi uchun kengaytirilgan matritsani yozishdir.
\[\left[ {\begin{array}{rr|r}1&{ - 7}&{ - 11}\\5&2&{ - 18}\end{array}} \right]\]2-qadam
Yuqori chap burchakdagi raqamni bitta qilishimiz kerak. Bunday holda, u allaqachon mavjud va shuning uchun ushbu muammo uchun ushbu bosqichda hech narsa qilish mumkin emas.
3-qadam
Keyinchalik, 1 dan pastdagi 5 ni nolga aylantirishimiz kerak va biz buni quyidagi elementar qator operatsiyasi bilan amalga oshirishimiz mumkin.
\[\left[ {\begin{array}{rr|r}1&{ - 7}&{ - 11}\\5&2&{ - 18}\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{{R_{\,2}} - 5{R_{\,1}} \to {R_{\,2}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rr|r}1&{ - 7}&{ - 11}\\0&{37}&{37}\end{array}} \right]\]4-qadam
Keyingi qadam, ikkinchi ustunning pastki qismidagi raqamni (bu holda 37) bittaga aylantirishdir. Quyidagi elementar qator operatsiyasi biz uchun buni amalga oshiradi.
\[\left[ {\begin{array}{rr|r}1&{ - 7}&{ - 11}\\0&{37}&{37}\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{\frac{1}{{37}}{R_{\,2}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rr|r}1&{ - 7}&{ - 11}\\0&1&1\end{array}} \right]\]5-qadam
Nihoyat, biz 4-bosqichda olingan raqamdan yuqoridagi raqamni nolga aylantirishimiz kerak. Buning uchun quyidagi elementar qator amalidan foydalanishimiz mumkin.
\[\left[ {\begin{array}{rr|r}1&{ - 7}&{ - 11}\\0&1&1\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{{R_{\,1}} + 7{R_{\,2}} \to {R_{\,1}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rr|r}1&0&{ - 4}\\0&1&1\end{array}} \right]\]6-qadam
Biz 5-bosqichda topilgan yakuniy kengaytirilgan matritsadan tizimning yechimini olamiz: \(\require{bbox} \bbox[2pt,border:1px solid black]{{x = - 4,\,\,y = 1}}\).