Matematică
misuadina
2016-04-14 03:39:48
Salut, ma puteți ajuta va rog?!
Răspunsuri la întrebare
djek
2016-04-14 06:59:27

[latex]displaystyle mathtt{a)~B= left(egin{array}{ccc}mathtt0&mathtt1&mathtt{-1}\mathtt2&mathtt3&mathtt0\mathtt1&mathtt{-1}&mathtt1end{array} ight)}\ \ mathtt{det(B)=left|egin{array}{ccc}mathtt0&mathtt1&mathtt{-1}\mathtt2&mathtt3&mathtt0\mathtt1&mathtt{-1}&mathtt1end{array} ight|=0cdot3cdot1+(-1)cdot2cdot(-1)+1cdot0cdot1-}\ \ mathtt{-(-1)cdot3cdot1-1cdot2cdot1-0cdot0cdot(-1)=3}\ \ mathtt{det(B)=3 e 0 Rightarrow Matricea~B~este~inversabilu{a}}[/latex] [latex]displaystyle mathtt{b)~A=left(egin{array}{ccc}mathtt1&mathtt1&mathtt2\mathtt0&mathtt1&mathtt3\mathtt0&mathtt0&mathtt1end{array} ight)~~~~~~~~~~~~~~~~~~~~~~~~~~~~A^{-1}= frac{1}{det(A)}cdot A^* }[/latex] [latex]displaystyle mathtt{det(A)=left|egin{array}{ccc}mathtt1&mathtt1&mathtt2\mathtt0&mathtt1&mathtt3\mathtt0&mathtt0&mathtt1end{array} ight|=1cdot1cdot1+2cdot0 cdot0+1cdot3cdot0-2cdot1cdot0-}\ \ mathtt{-1cdot0cdot1-1cdot3cdot0=1}\ \ mathtt{~~~~~~~~~~~~~~~~~~~~~~~~~~det(A)=1}[/latex] [latex]displaystyle mathtt{A=left(egin{array}{ccc}mathtt1&mathtt1&mathtt2\mathtt0&mathtt1&mathtt3\mathtt0&mathtt0&mathtt1end{array} ight)Rightarrow A^{tr}=left(egin{array}{ccc}mathtt1&mathtt0&mathtt0\mathtt1&mathtt1&mathtt0\mathtt2&mathtt3&mathtt1end{array} ight)}[/latex] [latex]displaystyle mathtt{D_{11}=(-1)^{1+1}cdot left|egin{array}{ccc}mathtt1&mathtt0\mathtt3&mathtt1end{array} ight|=1 cdot 1=1}\ \ mathtt{D_{12}=(-1)^{1+2}cdot left|egin{array}{ccc}mathtt1&mathtt0\mathtt2&mathtt1end{array} ight|=(-1) cdot 1=-1}\ \ mathtt{D_{13}=(-1)^{1+3}cdot left|egin{array}{ccc}mathtt1&mathtt1\mathtt2&mathtt3end{array} ight|=1 cdot 1=1}[/latex] [latex]mathtt{D_{21}=(-1)^{2+1}cdot left|egin{array}{ccc}mathtt0&mathtt0\mathtt3&mathtt1end{array} ight|=(-1)cdot 0=0}\ \ mathtt{D_{22}=(-1)^{2+2}cdot left|egin{array}{ccc}mathtt1&mathtt0\mathtt2&mathtt1end{array} ight|=1cdot1=1}\ \ mathtt{D_{23}=(-1)^{2+3}cdot left|egin{array}{ccc}mathtt1&mathtt0\mathtt2&mathtt3end{array} ight|=(-1)cdot3=-3}[/latex] [latex]displaystyle mathtt{D_{31}=(-1)^{3+1}cdot left|egin{array}{ccc}mathtt0&mathtt0\mathtt1&mathtt0end{array} ight|=1 cdot 0=0}\ \ mathtt{D_{32}=(-1)^{3+2}cdot left|egin{array}{ccc}mathtt1&mathtt0\mathtt1&mathtt0end{array} ight|=(-1)cdot 0=0}\ \ mathtt{D_{33}=(-1)^{3+3}cdot left|egin{array}{ccc}mathtt1&mathtt0\mathtt1&mathtt1end{array} ight|=1 cdot 1=1}[/latex] [latex]displaystyle mathtt{A^*= left(egin{array}{ccc}mathtt1&mathtt{-1}&mathtt1\mathtt0&mathtt1&mathtt{-3}\mathtt0&mathtt0&mathtt1end{array} ight)}\ \ mathtt{A^{-1}= frac{1}{1}cdot left(egin{array}{ccc}mathtt1&mathtt{-1}&mathtt1\mathtt0&mathtt1&mathtt{-3}\mathtt0&mathtt0&mathtt1end{array} ight)= left(egin{array}{ccc}mathtt1&mathtt{-1}&mathtt1\mathtt0&mathtt1&mathtt{-3}\mathtt0&mathtt0&mathtt1end{array} ight) }[/latex] [latex]displaystyle mathtt{c)~Xcdot A=BRightarrow X=BA^{-1}}\ \ mathtt{X=left(egin{array}{ccc}mathtt0&mathtt1&mathtt{-1}\mathtt2&mathtt3&mathtt0\mathtt1&mathtt{-1}&mathtt1end{array} ight)cdot left(egin{array}{ccc}mathtt1&mathtt{-1}&mathtt1\mathtt0&mathtt1&mathtt{-3}\mathtt0&mathtt0&mathtt1end{array} ight)} [/latex] [latex]displaystyle mathtt{X= left(egin{array}{ccc}mathtt{0cdot1+1cdot0+(-1)cdot0}&mathtt{0cdot(-1)+1cdot1+(-1)cdot0}&mathtt{0cdot1+1cdot(-3)+(-1)cdot1}\mathtt{2cdot1+3cdot0+0cdot0}&mathtt{2cdot(-1)+3cdot1+0cdot0}&mathtt{2cdot1+3cdot(-3)+0cdot1}\mathtt{1cdot1+(-1)cdot0+1cdot0}&mathtt{1cdot(-1)+(-1)cdot1+1cdot0}&mathtt{1cdot1+(-1)cdot(-3)+1cdot1}end{array} ight)} [/latex] [latex]displaystyle mathtt{X= left(egin{array}{ccc}mathtt{0+0+0}&mathtt{0+1+0}&mathtt{0-3-1}\mathtt{2+0+0}&mathtt{-2+3+0}&mathtt{2-9+0}\mathtt{1+0+0}&mathtt{-1-1+0}&mathtt{1+3+1}end{array} ight)}\ \ mathtt{X=left(egin{array}{ccc}mathtt0&mathtt1&mathtt{-4}\mathtt2&mathtt1&mathtt{-7}\mathtt1&mathtt{-2}&mathtt5end{array} ight)}[/latex]

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