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[tex]\displaystyle
\mathtt{\left\{\begin{array}{ccc}\mathtt{x+y-2z=0}\\\mathtt{x-y+z=1}\\\mathtt{x+y+az=2}\end{array}\right,~a\in\mathbb{R}~~~~~~~~~~~~~~~~~~~~~~~~A=
\left(\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt{-2}\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt1&\mathtt1&\mathtt
a\end{array}\right)}[/tex]
[tex]\displaystyle \mathtt{a)~det(A)=\left|\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt{-2}\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt1&\mathtt1&\mathtt a\end{array}\right|=1 \cdot (-1) \cdot a+(-2)\cdot1\cdot1+1\cdot1\cdot1-}\\ \\ \mathtt{-(-2)\cdot(-1)\cdot1-1\cdot1\cdot a-1\cdot1\cdot1=-a-2+1-2-a-1=}\\ \\ \mathtt{=-2a-4}}[/tex]
[tex]\displaystyle \mathtt{b)~Matricea~A~este~inversabil\u{a}~\Leftrightarrow~det(A)\ne 0}\\ \\ \mathtt{det(A)=-2a-4}\\ \\ \mathtt{-2a-4\ne0 \Rightarrow a\ne -2}\\ \\ \mathtt{Pentru~a\in\mathbb{R}-\{-2\}~matricea~A~este~inversabil\u{a}~.}[/tex]
[tex]\displaystyle \mathtt{c)~a=0\Rightarrow \mathtt{\left\{\begin{array}{ccc}\mathtt{x+y-2z=0}\\\mathtt{x-y+z=1}\\\mathtt{x+y=2}\end{array}\right}}[/tex]
[tex]\displaystyle \mathtt{\Delta= \left|\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt{-2}\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt1&\mathtt1&\mathtt 0\end{array}\right|=1 \cdot (-1) \cdot0+(-2)\cdot 1 \cdot 1+1\cdot1\cdot1-}\\ \\ \mathtt{-(-2)\cdot(-1)\cdot1-1 \cdot 1 \cdot0-1\cdot1\cdot1=-4}\\ \\ \mathtt{\Delta=-4\ne0}[/tex]
![\displaystyle \mathtt{\Delta_x= \left|\begin{array}{ccc}\mathtt0&\mathtt1&\mathtt{-2}\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt2&\mathtt1&\mathtt 0\end{array}\right|=0 \cdot (-1)\cdot0+(-2)\cdot 1\cdot1+1\cdot1\cdot2-}\\ \\ \mathtt{-(-2)\cdot (-1) \cdot 2-1 \cdot 1 \cdot 0-0 \cdot 1 \cdot 1=-4}\\ \\ \mathtt{\Delta_x=-4} \displaystyle \mathtt{\Delta_x= \left|\begin{array}{ccc}\mathtt0&\mathtt1&\mathtt{-2}\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt2&\mathtt1&\mathtt 0\end{array}\right|=0 \cdot (-1)\cdot0+(-2)\cdot 1\cdot1+1\cdot1\cdot2-}\\ \\ \mathtt{-(-2)\cdot (-1) \cdot 2-1 \cdot 1 \cdot 0-0 \cdot 1 \cdot 1=-4}\\ \\ \mathtt{\Delta_x=-4}](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cmathtt%7B%5CDelta_x%3D+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D%5Cmathtt0%26amp%3B%5Cmathtt1%26amp%3B%5Cmathtt%7B-2%7D%5C%5C%5Cmathtt1%26amp%3B%5Cmathtt%7B-1%7D%26amp%3B%5Cmathtt1%5C%5C%5Cmathtt2%26amp%3B%5Cmathtt1%26amp%3B%5Cmathtt+0%5Cend%7Barray%7D%5Cright%7C%3D0+%5Ccdot+%28-1%29%5Ccdot0%2B%28-2%29%5Ccdot+1%5Ccdot1%2B1%5Ccdot1%5Ccdot2-%7D%5C%5C+%5C%5C+%5Cmathtt%7B-%28-2%29%5Ccdot+%28-1%29+%5Ccdot+2-1+%5Ccdot+1+%5Ccdot+0-0+%5Ccdot+1+%5Ccdot+1%3D-4%7D%5C%5C+%5C%5C+%5Cmathtt%7B%5CDelta_x%3D-4%7D)
![\displaystyle \mathtt{\Delta_y= \left|\begin{array}{ccc}\mathtt1&\mathtt0&\mathtt{-2}\\\mathtt1&\mathtt1&\mathtt1\\\mathtt1&\mathtt2&\mathtt 0\end{array}\right|=1 \cdot 1 \cdot 0+(-2)\cdot 1\cdot2+0 \cdot 1 \cdot 1-(-2)\cdot1\cdot1-}\\ \\ \mathtt{-0 \cdot 1 \cdot 0-1 \cdot 1 \cdot 2=-4}\\ \\ \mathtt{\Delta_y=-4} \displaystyle \mathtt{\Delta_y= \left|\begin{array}{ccc}\mathtt1&\mathtt0&\mathtt{-2}\\\mathtt1&\mathtt1&\mathtt1\\\mathtt1&\mathtt2&\mathtt 0\end{array}\right|=1 \cdot 1 \cdot 0+(-2)\cdot 1\cdot2+0 \cdot 1 \cdot 1-(-2)\cdot1\cdot1-}\\ \\ \mathtt{-0 \cdot 1 \cdot 0-1 \cdot 1 \cdot 2=-4}\\ \\ \mathtt{\Delta_y=-4}](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cmathtt%7B%5CDelta_y%3D+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D%5Cmathtt1%26amp%3B%5Cmathtt0%26amp%3B%5Cmathtt%7B-2%7D%5C%5C%5Cmathtt1%26amp%3B%5Cmathtt1%26amp%3B%5Cmathtt1%5C%5C%5Cmathtt1%26amp%3B%5Cmathtt2%26amp%3B%5Cmathtt+0%5Cend%7Barray%7D%5Cright%7C%3D1+%5Ccdot+1+%5Ccdot+0%2B%28-2%29%5Ccdot+1%5Ccdot2%2B0+%5Ccdot+1+%5Ccdot+1-%28-2%29%5Ccdot1%5Ccdot1-%7D%5C%5C+%5C%5C+%5Cmathtt%7B-0+%5Ccdot+1+%5Ccdot+0-1+%5Ccdot+1+%5Ccdot+2%3D-4%7D%5C%5C+%5C%5C+%5Cmathtt%7B%5CDelta_y%3D-4%7D)
![\displaystyle \mathtt{\Delta_z= \left|\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt0\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt1&\mathtt1&\mathtt 2\end{array}\right|=1 \cdot (-1) \cdot 2+0 \cdot 1 \cdot 1+1 \cdot 1 \cdot 1-0 \cdot (-1) \cdot 1-}\\ \\ \mathtt{-1 \cdot 1 \cdot 2-1 \cdot 1 \cdot 1=-4}\\ \\ \mathtt{\Delta_z=-4} \displaystyle \mathtt{\Delta_z= \left|\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt0\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt1&\mathtt1&\mathtt 2\end{array}\right|=1 \cdot (-1) \cdot 2+0 \cdot 1 \cdot 1+1 \cdot 1 \cdot 1-0 \cdot (-1) \cdot 1-}\\ \\ \mathtt{-1 \cdot 1 \cdot 2-1 \cdot 1 \cdot 1=-4}\\ \\ \mathtt{\Delta_z=-4}](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cmathtt%7B%5CDelta_z%3D+%5Cleft%7C%5Cbegin%7Barray%7D%7Bccc%7D%5Cmathtt1%26amp%3B%5Cmathtt1%26amp%3B%5Cmathtt0%5C%5C%5Cmathtt1%26amp%3B%5Cmathtt%7B-1%7D%26amp%3B%5Cmathtt1%5C%5C%5Cmathtt1%26amp%3B%5Cmathtt1%26amp%3B%5Cmathtt+2%5Cend%7Barray%7D%5Cright%7C%3D1+%5Ccdot+%28-1%29+%5Ccdot+2%2B0+%5Ccdot+1+%5Ccdot+1%2B1+%5Ccdot+1+%5Ccdot+1-0+%5Ccdot+%28-1%29+%5Ccdot+1-%7D%5C%5C+%5C%5C+%5Cmathtt%7B-1+%5Ccdot+1+%5Ccdot+2-1+%5Ccdot+1+%5Ccdot+1%3D-4%7D%5C%5C+%5C%5C++%5Cmathtt%7B%5CDelta_z%3D-4%7D)
[tex]\displaystyle \mathtt{a)~det(A)=\left|\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt{-2}\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt1&\mathtt1&\mathtt a\end{array}\right|=1 \cdot (-1) \cdot a+(-2)\cdot1\cdot1+1\cdot1\cdot1-}\\ \\ \mathtt{-(-2)\cdot(-1)\cdot1-1\cdot1\cdot a-1\cdot1\cdot1=-a-2+1-2-a-1=}\\ \\ \mathtt{=-2a-4}}[/tex]
[tex]\displaystyle \mathtt{b)~Matricea~A~este~inversabil\u{a}~\Leftrightarrow~det(A)\ne 0}\\ \\ \mathtt{det(A)=-2a-4}\\ \\ \mathtt{-2a-4\ne0 \Rightarrow a\ne -2}\\ \\ \mathtt{Pentru~a\in\mathbb{R}-\{-2\}~matricea~A~este~inversabil\u{a}~.}[/tex]
[tex]\displaystyle \mathtt{c)~a=0\Rightarrow \mathtt{\left\{\begin{array}{ccc}\mathtt{x+y-2z=0}\\\mathtt{x-y+z=1}\\\mathtt{x+y=2}\end{array}\right}}[/tex]
[tex]\displaystyle \mathtt{\Delta= \left|\begin{array}{ccc}\mathtt1&\mathtt1&\mathtt{-2}\\\mathtt1&\mathtt{-1}&\mathtt1\\\mathtt1&\mathtt1&\mathtt 0\end{array}\right|=1 \cdot (-1) \cdot0+(-2)\cdot 1 \cdot 1+1\cdot1\cdot1-}\\ \\ \mathtt{-(-2)\cdot(-1)\cdot1-1 \cdot 1 \cdot0-1\cdot1\cdot1=-4}\\ \\ \mathtt{\Delta=-4\ne0}[/tex]
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