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Matrice ¸si <strong>de</strong>terminant¸i 13<br />
Observat¸ia 1.3 In cazul în care sistemul <strong>de</strong> trei ecuat¸ii cu trei necunoscute<br />
⎧<br />
⎪⎨ α11 x1 + α12 x2 + α13 x3 = β1<br />
⎪⎩<br />
α21 x1 + α22 x2 + α23 x3 = β2<br />
α31 x1 + α32 x2 + α33 x3 = β3<br />
are solut¸ie unică, solut¸ia obt¸inută prin metoda reducerii se poate scrie<br />
<br />
<br />
β1<br />
<br />
β2<br />
<br />
β3<br />
x1 = <br />
<br />
α11<br />
<br />
α21<br />
<br />
α31<br />
α12<br />
α22<br />
α32<br />
α12<br />
α22<br />
α32<br />
<br />
<br />
α13 <br />
<br />
α23 <br />
<br />
α33 <br />
<br />
<br />
,<br />
α13 <br />
<br />
α23 <br />
<br />
α33 <br />
<br />
<br />
α11<br />
<br />
α21<br />
<br />
α31<br />
x2 = <br />
<br />
α11<br />
<br />
α21<br />
<br />
α31<br />
β1<br />
β2<br />
β3<br />
α12<br />
α22<br />
α32<br />
<br />
<br />
α13 <br />
<br />
α23 <br />
<br />
α33 <br />
<br />
<br />
,<br />
α13 <br />
<br />
α23 <br />
<br />
α33 <br />
<br />
<br />
α11<br />
<br />
α21<br />
<br />
α31<br />
x3 = <br />
<br />
α11<br />
<br />
α21<br />
<br />
α31<br />
α12<br />
α22<br />
α32<br />
α12<br />
α22<br />
α32<br />
<br />
<br />
β1 <br />
<br />
β2 <br />
<br />
β3 <br />
<br />
<br />
α13 <br />
<br />
α23 <br />
<br />
α33 <br />
dacă se utilizează notat¸ia<br />
<br />
<br />
<br />
<br />
a11 a12 a13 <br />
<br />
<br />
a21 a22 a23 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32<br />
<br />
<br />
a31 a32 a33 <br />
−a13 a22 a31 − a11 a23 a32 − a12 a21 a33.<br />
Definit¸ia 1.11 Fie K unul dintre corpurile R, C ¸si fie matricea pătrată<br />
⎛<br />
⎞<br />
Numărul<br />
<br />
<br />
<br />
<br />
<strong>de</strong>t A = <br />
<br />
<br />
A =<br />
⎜<br />
⎝<br />
a11 a12 a13<br />
a21 a22 a23<br />
a31 a32 a33<br />
a11 a12 a13<br />
a21 a22 a23<br />
a31 a32 a33<br />
se nume¸ste <strong>de</strong>terminantul matricei A.<br />
⎟<br />
⎠ ∈ M3×3(K).<br />
<br />
<br />
<br />
<br />
= a11 a22 a33 + a12 a23 a31 + a13 a21 a32<br />
<br />
<br />
−a13 a22 a31 − a11 a23 a32 − a12 a21 a33<br />
Definit¸ia 1.12 Prin permutare <strong>de</strong> grad n se înt¸elege o funct¸ie bijectivă<br />
σ : {1, 2, ..., n} −→ {1, 2, ..., n}.<br />
(1.2)
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