Question

In special, punctul C), dar as aprecia și dacă m ați ajuta cu punctul B)
Va rog frumos!! Dau coroana

Answer 1

$\displaystyle 1.~~~\left\{\begin{array}{ccc}x-y+2z=a\\2x-y=0\\y-z=1\end{array}\right ~~~~~~~~~~~~~~~~~~~~~~~~A= \left(\begin{array}{ccc}1&-1&2\\2&-1&0\\0&1&-1\end{array}\right)\\ \\ \\ b)~det~A=\left|\begin{array}{ccc}1&-1&2\\2&-1&0\\0&1&-1\end{array}\right|=1\cdot(-1)\cdot(-1)+2\cdot2\cdot1+(-1)\cdot0\cdot0-\\ \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-2\cdot(-1)\cdot0-(-1)\cdot2\cdot(-1)-(-1)\cdot0\cdot1=\\ \\ =1+4+0-0-2-0=3\\ \\ det~A=3$

$\displaystyle c)~a=-2\Rightarrow \left\{\begin{array}{ccc}x-y+2z=-2\\2x-y=0\\y-z=1\end{array}\right\\ \\ \\ \Delta=\left|\begin{array}{ccc}1&-1&2\\2&-1&0\\0&1&-1\end{array}\right|=1\cdot(-1)\cdot(-1)+2\cdot2\cdot1+(-1)\cdot0\cdot0-\\ \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-2\cdot(-1)\cdot0-(-1)\cdot2\cdot(-1)-(-1)\cdot0\cdot1=\\ \\ =1+4+0-0-2-0=3 \\ \\ \Delta=3\ne0$

$\displaystyle \Delta_x=\left|\begin{array}{ccc}-2&-1&2\\0&-1&0\\1&1&-1\end{array}\right|=(-2)\cdot(-1)\cdot(-1)+2\cdot0\cdot1+(-1)\cdot0\cdot1-\\ \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-~2\cdot(-1)\cdot1-(-1)\cdot0\cdot(-1)-(-2)\cdot0\cdot1=\\ \\ =-2+0+0+2-0-0=0\\ \\ \Delta_x=0$

$\displaystyle \Delta_y=\left|\begin{array}{ccc}1&-2&2\\2&0&0\\0&1&-1\end{array}\right|=1\cdot0\cdot(-1)+2\cdot2\cdot1+1\cdot0\cdot1-2\cdot0\cdot 0-\\ \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-(-2)\cdot 2\cdot(-1)-1\cdot0\cdot1=\\ \\ =0+4+0-0-4-0=0\\ \\ \Delta_y=0$

$\displaystyle \Delta_z=\left|\begin{array}{ccc}1&-1&-2\\2&-1&0\\0&1&1\end{array}\right|=1\cdot(-1)\cdot1+(-2)\cdot2\cdot1+(-1)\cdot0\cdot0-\\ \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-(-2)\cdot(-1)\cdot0-(-1)\cdot2\cdot1-1\cdot0\cdot1=\\ \\ =-1-4+0-0+2-0=-3\\ \\ \Delta_z=-3\\ \\ x= \frac{\Delta_x}{\Delta}= \frac{0}{3} =0~~~~~~~~~~~~~~~~~~~~~~~~~x=0\\ \\ y= \frac{\Delta_y}{\Delta}= \frac{0}{3} =0~~~~~~~~~~~~~~~~~~~~~~~~~y=0\\ \\ z= \frac{\Delta_z}{\Delta}= \frac{-3}{3} =-1 ~~~~~~~~~~~~~~~~~~~~~z=-1$