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Matritsa ustida satr amallari

Matritsa ustida sodda satr amallaridan foydalanishni bilib oling. Bu bizga murakkab chiziqli tenglamalar sistemasini yechishga yordam beradi.

Matritsa satrlari ustida amallar

Quyidagi jadval matritsa satrlari ustida amallarni keltiradi.
Matrisa satrlari ustida amallarMasala
Ixtiyoriy ikki satrni almashtiring[253346][346253]  (1- va 2-satrlar oʻrnini almashtiring.)\left[\begin{array}{rr}{\blueD2} & {\blueD5} &{ \blueD{3}} \\ \greenD{3} &\greenD {4} &\greenD {6} \end{array}\right]\rightarrow\left[\begin{array}{rr} \greenD{3} & \greenD{4} &\greenD {6}\\\blueD{2} &\blueD {5} &\blueD{ 3} \end{array}\right]\\\\~~\\ \\ {\text{(1- va 2-satrlar oʻrnini almashtiring.)}}
Satrni 0 boʻlmagan songa koʻpaytiring[253346][323533346] (Satrning 3 karra koʻpaytmasi.)\left[\begin{array}{rr}{\maroonD2} & {\maroonD5} &{ \maroonD3} \\ {3} & {4} & {6} \end{array}\right]\rightarrow\left[\begin{array}{rrr}{\goldD3 \cdot \maroonD2} & {\goldD3 \cdot \maroonD5} &{ \goldD3 \cdot \maroonD3} \\ { 3} & { 4} & { 6} \end{array}\right] \\~\\ {\text{(Satrning 3 karra koʻpaytmasi.)}}
Bir satrni boshqasiga qoʻshing[253346][2533+24+56+3]  (Satr 1- va 2-satrlarning yigʻindisi boʻladi.)\left[\begin{array}{rr}{\tealD2} &\tealD5 &{ \tealD{3}} \\ \purpleC{3} &\purpleC {4} &\purpleC {6} \end{array}\right]\rightarrow\left[\begin{array}{rrr} {\tealD2} &\tealD5&{\tealD3}\\\purpleC{3}+\tealD2 & \purpleC{4}+\tealD5 &\purpleC{6} +\tealD3\end{array}\right]\\~~\\ {\text{(Satr 1- va 2-satrlarning yigʻindisi boʻladi.)}}
Satrlar ustida amallar tenglamalar sistemasini yechishda yordam beradi. Shu amallar haqidagi bilimlarimizni misollar orqali mustahkamlaylik.

Ixtiyoriy ikki satr oʻrinlarini almashtirish

Misol

Ushbu matritsa ustida R, start subscript, 1, end subscript, \leftrightarrow, R, start subscript, 2, end subscript satrlar oʻrnini almashtirish amalini bajaring.
[483245712]\left[\begin{array} {rrr} 4 & 8 & 3 \\ 2 & 4 & 5 \\ 7 & 1 & 2 \end{array} \right]

Yechim

R, start subscript, start color #11accd, 1, end color #11accd, end subscript, \leftrightarrow, R, start subscript, start color #1fab54, 2, end color #1fab54, end subscript amal start color #11accd, 1, end color #11accd, minus va start color #1fab54, 2, end color #1fab54, minussatrlar oʻrnini almashtirishni bildiradi.
Demak, [483245712]\left[\begin{array} {rrr} \blueD4 & \blueD8 & \blueD{3} \\ \greenD2 & \greenD4 & \greenD5 \\ 7 & 1 & 2 \end{array} \right] matritsa [245483712]\left[\begin{array} {rrr} \greenD2 & \greenD4 & \greenD5 \\ \blueD4 & \blueD8 & \blueD{3} \\ 7 & 1 & 2 \end{array} \right] matritsaga oʻzgaradi.
Siz bazida quyidagicha amal ifodalanishiga duch kelishingiz mumkin.
[483245712]R1R2[245483712]\left[\begin{array} {rrr} 4 & 8 & 3 \\ 2 & 4 & 5 \\ 7 & 1 & 2 \end{array} \right] \xrightarrow{{R_1\leftrightarrow R_2}}\left[\begin{array} {rrr} 2 & 4 & 5 \\ 4 &8 & 3 \\ 7 & 1 & 2 \end{array} \right]
1-satr 2-satr bilan va 2-satr 1-satr bilan oʻrin almashishiga va 3-satr oʻzgarmay qolganiga eʼtibor bering.
1-masala
  • Joriy
Ushbu matritsa ustida R, start subscript, 2, end subscript, \leftrightarrow, R, start subscript, 3, end subscript satr amalini bajaring.
[7296411312]\left[\begin{array} {rrr} 7 & 2 & 9 \\ 6 & 4 & 1 \\ 1 & 3 & 12 \end{array} \right]

Satrni 0 boʻlmagan songa koʻpaytirish

Misol

Ushbu matritsa ustida 3, R, start subscript, 2, end subscript, right arrow, R, start subscript, 2, end subscript amalni bajaring.
[661230459]\left[\begin{array} {rrr} 6 & 6 & 1 \\ 2 & 3 & 0 \\ 4 & 5 & 9 \end{array} \right]

Yechim

start color #ca337c, 3, end color #ca337c, R, start subscript, start color #e07d10, 2, end color #e07d10, end subscript, right arrow, R, start subscript, start color #e07d10, 2, end color #e07d10, end subscript satr amali start color #e07d10, 2, start text, negative, end text, end color #e07d10satrni start color #ca337c, 3, end color #ca337c marta orttirishni bildiradi.
[661230459]\left[\begin{array} {rrr} 6 & 6 & 1 \\ \goldD{2} & \goldD{3} & \goldD{0} \\ 4 & 5 & 9 \end{array} \right] matritsa[661323330459]=[661690459]\left[\begin{array} {rrr} 6 & 6 & 1 \\ \maroonD{3}\cdot \goldD{2} &\maroonD{3}\cdot \goldD{3} &\maroonD{3}\cdot \goldD{0} \\ 4 & 5 & 9 \end{array} \right] =\left[\begin{array} {rrr} 6 & 6 & 1 \\ 6 & 9 & {0} \\ 4 & 5 & 9 \end{array} \right] matritsaga oʻzgaradi.
Bu amal quyidagicha ifodalanadi:
[661230459]3R2R2[661690459]\left[\begin{array} {rrr} 6 & 6 & 1 \\ 2 & 3 & 0 \\ 4 & 5 & 9 \end{array} \right] \xrightarrow{3R_2\rightarrow R_2}\left[\begin{array} {rrr} 6 & 6 & 1 \\ 6 & 9 & {0} \\ 4 & 5 & 9 \end{array} \right]
2-satr 3 marta ortishiga va boshqa satrlar oʻzgarmay qolishiga eʼtibor bering.
3-misol
  • Joriy
Ushbu matritsa ustida 2, R, start subscript, 1, end subscript, right arrow, R, start subscript, 1, end subscript satr amalini bajaring.
[26517480]\left[\begin{array} {ccc} 2 & 6 & 5 & 1 \\ 7 & 4 & 8 & 0 \end{array} \right]

Bir satrni boshqasiga qoʻshish

Misol

Ushbu matritsa ustida R, start subscript, 1, end subscript, plus, R, start subscript, 2, end subscript, right arrow, R, start subscript, 2, end subscript amalni bajaring.
[234081]\left[\begin{array} {rrr} 2 & 3 & 4\\ 0 & 8 & 1 \end{array} \right]

Yechim

R, start subscript, start color #01a995, 1, end color #01a995, end subscript, plus, R, start subscript, start color #aa87ff, 2, end color #aa87ff, end subscript, right arrow, R, start subscript, 2, end subscript amal 2, start text, negative, end textsatrni start color #01a995, 1, start text, negative, end text, end color #01a995 va start color #aa87ff, 2, start text, negative, end text, end color #aa87ffsatrlar yigʻindisi bilan almashtirishni bildiradi.
[234081]\left[\begin{array} {rrr} \tealD2 & \tealD{3} &\tealD{ 4}\\ \purpleC0 & \purpleC8 & \purpleC1 \end{array} \right] matritsa[2342+03+84+1]=[2342115]\left[\begin{array} {lll} \tealD2 &{\tealD3} &{ \tealD4}\\ \tealD2+\purpleC0 & \tealD3+\purpleC8 & \tealD4 +\purpleC1 \end{array} \right]= \left[\begin{array} {rrr} 2 & 3 & 4\\ 2 & 11 & 5 \end{array} \right] matritsaga oʻzgaradi.
Bu amal quyidagicha ifodalanadi:
[234081]R1+R2R2[2342115]\left[\begin{array} {rrr} 2 & 3 & 4\\ 0 & 8 & 1 \end{array} \right] \xrightarrow{R_1+R_2\rightarrow R_2} \left[\begin{array} {rrr} 2 & 3 & 4\\ 2 & 11 & 5 \end{array} \right]
1- va 2-satrlar yigʻindisi 2-satr bilan almashishiga va qolgan satrlar oʻzgarmay qolishiga eʼtibor bering.
5-masala
  • Joriy
Ushbu matritsa ustida R, start subscript, 1, end subscript, plus, R, start subscript, 3, end subscript, right arrow, R, start subscript, 3, end subscript amalni bajaring.
[162350721]\left[\begin{array} {rrr} -1 & 6 & -2 \\ -3 & 5 & 0 \\ 7 & 2 & 1 \end{array} \right]

Murakkab masala
Ushbu matritsa ustida R, start subscript, 1, end subscript, plus, 2, R, start subscript, 3, end subscript, right arrow, R, start subscript, 1, end subscript satr amalini bajaring.
[573214886]\left[\begin{array} {rrr} -5 & 7 & 3 \\ -2 & -1 & 4 \\ 8 & 8 & -6 \end{array} \right]

Tenglamalar sistemasi va matritsa satrlari ustida amallar

Yodingizda boʻlsa, kengaytirilgan matritsaning har bir satri tenglamlar sistemasining bir tenglamasini ifodalar edi.
Masalan, chapdagi sistemaning matritsaviy ifodasi oʻngdagi matritsaga mos keladi.
SistemaMatritsa
1x+3y=52x+5y=6\begin{aligned} 1x+3y &=5\\2x+5y &=6\end{aligned}[135256]\left[\begin{array}{cc:c}1&3&5\\\\2&5&6\end{array}\right]
Kengaytirilgan matritsalar bilan ishlayotganda ekvivalent kengaytirilgan matritsa hosil qilish uchun ixtiyoriy satr amalidan foydalanishimiz mumkin. Bundan maqsadni quyida koʻrib chiqamiz:

Ixtiyoriy ikki satrni almashtirish

Ekvivalent sistemalarKengaytirilgan matritsa
1x+3y=52x+5y=6\begin{aligned} \blueD1x+\blueD3y &=\blueD{5} \\\greenD{2}x+\greenD{{5}}y &=\greenD{6} \end{aligned} [135256]\left[\begin{array}{cc:c}\blueD1&\blueD3&\blueD5\\\\\greenD2&\greenD5&\greenD6\end{array}\right]
\downarrow
2x+5y=61x+3y=5\begin{aligned}\greenD{2}x+\greenD{{5}}y &=\greenD{6}\\ \blueD1x+\blueD3y &=\blueD{5} \end{aligned}[256135]\left[\begin{array}{cc:c}\greenD2&\greenD5&\greenD6\\\\\blueD1&\blueD3&\blueD5\end{array}\right]
Yuqoridagi tenglamalar sistemalari ekvivalent hisoblanadi. Chunki sistemada tenglamalar tartibi ahamiyatsiz. Shunday ekan, matritsada ham ixtiyoriy ikki satrning oʻzaro oʻrinlarini almashtirish mumkin.

Satrni 0 boʻlmagan songa koʻpaytirish

Tenglamaning ikki tarafini 0 boʻlmagan ixtiyoriy songa koʻpaytirish bizga ekvivalent tenglama beradi.
Tenglamalar sistemasini yechishda nomaʼlumni yoʻqotish uchun bu amaldan koʻp foydalanamiz. Chunki ikki tenglama ekvivalent boʻlsa, ikki sistema ham ekvivalent boʻladi.
Ekvivalent sistemalarKengaytirilgan matritsa
1x+3y=52x+5y=6\begin{aligned} \maroonD1x+\maroonD3y &=\maroonD5 \\2x+5y &=6\end{aligned} [135256]\left[\begin{array}{cc:c}\maroonD1 & \maroonD3 &\maroonD5 \\2&5&6\end{array}\right]
\downarrow
2x+(6)y=102x+()5y=6\begin{aligned}\goldD{-2}x+(\goldD{-6})y &=\goldD{-10} \\2x+\phantom{(-)}5y &=6\end{aligned} [2610256]\left[\begin{array}{rr:r}\goldD{-2}&\goldD{-6}& \goldD{-10}\\2&5&6\end{array}\right]
Demak, kengaytirilgan matritsadan tenglamalar sistemasini yechishda foydalanayotgan boʻlsak, uning ixtiyoriy satrini 0 boʻlmagan songa koʻpaytirishimiz mumkin.

Bir satrni boshqasiga qoʻshish

Bilamizki, tenglamaning ikki tarafiga bir xil qiymat qoʻshilsa, ekvivalent boʻlgan tenglama hosil boʻladi.
Demak, agar A, equals, B va C, equals, D boʻlsa, A, plus, C, equals, B, plus, D boʻladi.
Biz bundan tenglamalar sistemasini yechishda keng foydalanamiz. Masalan, bu sistemada 2x6y=102x+5y=6\begin{aligned}-2x-6y &=-10 \\ {2}x+{{5}}y &={6}\end{aligned} tenglamalarni qoʻshib, ushbu ifodaga erishishimiz mumkin: minus, y, equals, minus, 4.
Endi bu ifodani yuqorida biror tenglama bilan sistemaga qoʻysak, yangi ekvivalent sistema hosil boʻladi.
Ekvivalent sistemalarKengaytirilgan matritsa
2x6y=102x+5y=6\begin{aligned} -2x-6y &=-10\\2x+5y &=6\end{aligned} [2610256]\left[\begin{array}{rr:r}-2&-6&-10\\2&5&6\end{array}\right]
\downarrow
2x+(6)y=100x+(1)y=4\begin{aligned}-2x+(-6)y &=-10\\\purpleC0x+(\purpleC{-1})y &=\purpleC{-4} \end{aligned}[2610014]\left[\begin{array}{rr:r}-2&-6&-10\\\purpleC0&\purpleC{-1}&\purpleC{-4}\end{array}\right]
Demak, kengaytirilgan matritsadan tenglamalar sistemasini yechishda foydalanayotgan boʻlsak, ixtiyoriy bir satrni boshqasiga qoʻshishimiz mumkin.
Yakuniy murakkabroq masala
Satr amallari ketma-ketligi ushbu [2210233]\left[\begin{array}{rrr}{2} & {2} &{ 10} \\ {-2} & {-3} & {3} \end{array}\right] matritsada amalga oshirildi. Quyida har bir ketma-ketlik natijasi bilan tanishishingiz mumkin.
Amallarni ketma-ket bajaring.
Berilgan matritsa: [2210233]\left[\begin{array}{rr:r}2&2&10\\-2 & -3 & 3\end{array}\right]
1

Berilgan matritsa ushbu 2x+2y=102x3y=3\begin{aligned} 2x+2y &={10} \\ {-2}x-3y &={ 3} \end{aligned} sistemaga, natijaviy matritsamiz esa x=18y=13\begin{aligned} x&=18 \\ y&=-13 \end{aligned} sistemaga, yaʼni natijaga mos kelishiga eʼtibor bering.
Tenglamalar sistemasi faqatgina kengaytirilgan matritsa va satr amallaridan foydalanib yechildi.