Prezentare determinanti

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NeculaiDaniela

determinanti

Proprietatea 1.
 Determinantul unei matrice coincide cu
determinantul matricei transpuse
 Exemplu:
214
121
001
−
−
=
210
120
411
−
−

Proprietatea 2.
 Daca toate elementele unei linii (sau
coloane) dintr-o matrice sunt nule , atunci
determinantul matricie este nul.
 Exemplu:
743
000
121 −
=0

Proprietatea 3.
 Daca intr-o matrice schimbam doua linii (sau
coloane) intre ele obtinem o matrice care are
determinantul egal cu opusul determinantului
matricei initiale.
 Exemplu:
423
112
710
−− = -
423
710
112 −−

Proprietatea 4.
 Daca o martce are doua linii (sau coloane)
identice , atunci determinantul sau este nul.
 Exemplu:
424
262
151
−
−−
= 0

Proprietate 5.
 Daca toate elementele unei linii (sau
coloane) ale unei matrice sunt inmultite cu un
numar k obtinem o matrice al carei
determinant este egal cu determinantul
matricei initiale inmultit cu k.
 Exemplu:
124
262
134
−
−−
−
= 2
124
131
134
−
−−
−

Proprietatea 6.
Daca elementele a doua linii (sau coloane) ale
unei matrice sunt proportionale , atunci
determinantul matricei este nul.
Exemplu:
512
973
1024
− = 0

Proprietatea 7.
 Fie A=(aij)1<=i,j<=n o matrice patratica de ordinul
n. Presupunem ca elementele liniei i sunt de
forma aij=bij+cij ,oricare ar fi j=1,2,…,n.
Daca B (respectiv C) este matricea care se
obtine din A inlocuind elementele de pe linia i
cu elementele bij (respectiv cij ), j=1,2,…,n
,atunci
det(A)=det(B)+det(C).
 Exemplu:
152
666
421
−−
=
152
222
421
−−
+
152
444
421
−−

Proprietatea 8.
 Daca o linie (sau coloana) a unei matrici
patratice este combinatie liniara de celelalte
linii (sau coloane) , atunci determinantul
matricei este 0.

Proprietatea 9.
 Daca la o linie (sau coloana) a matricei A
adunam elementele altei linii (sau coloane)
inmultite cu acelasi numar , atunci aceasta
matrice are acelasi determinant ca si
matricea A.
 Exercitiu:
Sa se calculeze determinantul:
1130
1514
2112
6331
−
−
−
−

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Prezentare determinanti

2. Proprietatea 1.  Determinantul unei matrice coincide cu determinantul matricei transpuse  Exemplu: 214 121 001 − − = 210 120 411 − −

3. Proprietatea 2.  Daca toate elementele unei linii (sau coloane) dintr-o matrice sunt nule , atunci determinantul matricie este nul.  Exemplu: 743 000 121 − =0

4. Proprietatea 3.  Daca intr-o matrice schimbam doua linii (sau coloane) intre ele obtinem o matrice care are determinantul egal cu opusul determinantului matricei initiale.  Exemplu: 423 112 710 −− = - 423 710 112 −−

5. Proprietatea 4.  Daca o martce are doua linii (sau coloane) identice , atunci determinantul sau este nul.  Exemplu: 424 262 151 − −− = 0

6. Proprietate 5.  Daca toate elementele unei linii (sau coloane) ale unei matrice sunt inmultite cu un numar k obtinem o matrice al carei determinant este egal cu determinantul matricei initiale inmultit cu k.  Exemplu: 124 262 134 − −− − = 2 124 131 134 − −− −

7. Proprietatea 6. Daca elementele a doua linii (sau coloane) ale unei matrice sunt proportionale , atunci determinantul matricei este nul. Exemplu: 512 973 1024 − = 0

8. Proprietatea 7.  Fie A=(aij)1<=i,j<=n o matrice patratica de ordinul n. Presupunem ca elementele liniei i sunt de forma aij=bij+cij ,oricare ar fi j=1,2,…,n. Daca B (respectiv C) este matricea care se obtine din A inlocuind elementele de pe linia i cu elementele bij (respectiv cij ), j=1,2,…,n ,atunci det(A)=det(B)+det(C).  Exemplu: 152 666 421 −− = 152 222 421 −− + 152 444 421 −−

9. Proprietatea 8.  Daca o linie (sau coloana) a unei matrici patratice este combinatie liniara de celelalte linii (sau coloane) , atunci determinantul matricei este 0.

10. Proprietatea 9.  Daca la o linie (sau coloana) a matricei A adunam elementele altei linii (sau coloane) inmultite cu acelasi numar , atunci aceasta matrice are acelasi determinant ca si matricea A.  Exercitiu: Sa se calculeze determinantul: 1130 1514 2112 6331 − − − −

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Prezentare determinanti

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Prezentare determinanti