Yechishning matritsa, Gauss va Gauss-Jordan usullari

A  ^
21 22 2n _{, X  }
2 _{,
B }  ² 

^ ... ... ... ... ^{    }

^ a a
...
_a   _x   _b 

 n1 n2
nn   n  
n  _.

Bu yerda, ^{A } noma’lumlar oldida turgan koeffitsiyentlardan tuzilgan matritsa; ^{X }noma’lumlardan tuzilgan matritsa; ^{B } ozod hadlardan tuzilgan matritsa. U holda (1) tenglamalar sistemasini

koʻrinishda ifodalash mumkin.
AX  B
(2)

Faraz qilamiz, ^det
A  0
boʻlsin. U holda ^A matritsa uchun
^A1 teskari matritsa

mavjud. ^{AX
 B} tenglikning har ikkala tomonini
_A1
ga chapdan koʻpaytiramiz:

A^1AX  A^1B,
EX  A^1B,
X  A^1B.

Hosil boʻlgan
X  A^1B
ifoda chiziqli tenglamalar sistemasini matritsalar usuli

bilan yechish formulasidan iborat.
1-misol. Chiziqli tenglamalar sistemasini matritsalar usuli bilan yeching:
_2x₁  2x₂  3x₃  5,
^ x  x  x  0,
^ 1 2 3
^ 3x  x  x  2.
 1 2 3

Yechish.
^{A, X , B} matritsalarni tuzib olamiz:

^ 2 2
3 ^
 _x1 
 ₅ 

     

A  _ 1 1 1 _
X  _ x_{2 }
B  _ 0 _

^ 3 1 1 ^
 _x 
 ₂ 

  _,
 ³  _,
  _.

Bundan, ^{det A  12  0.} Teskari matritsani topamiz:
1 1 1 1 1 1

A₁₁ 
 2
1 1 _,
A₁₂   _{3 1}  2,
A₁₃  _{3 1}
 4,

2 3 2 3 2 2
A₂₁   _{1 1}  5, A₂₂  _{3 1}  11, A₂₃   _{3 1}  4,

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