Question

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Answer 1

$\displaystyle \mathtt{1)~A= \left(\begin{array}{ccc}\mathtt{-1}&\mathtt2&\mathtt0\\\mathtt1&\mathtt0&\mathtt1\\\mathtt{-2}&\mathtt1&\mathtt{-4}\end{array}\right);~I_3=\left(\begin{array}{ccc}\mathtt{1}&\mathtt0&\mathtt0\\\mathtt0&\mathtt1&\mathtt0\\\mathtt0&\mathtt0&\mathtt1\end{array}\right)~~~~~~~~~~~~~~A+3I_3=?}$

$\displaystyle \mathtt{3I_3=3 \cdot \left(\begin{array}{ccc}\mathtt{1}&\mathtt0&\mathtt0\\\mathtt0&\mathtt1&\mathtt0\\\mathtt0&\mathtt0&\mathtt1\end{array}\right)=\left(\begin{array}{ccc}\mathtt{3}&\mathtt0&\mathtt0\\\mathtt0&\mathtt3&\mathtt0\\\mathtt0&\mathtt0&\mathtt3\end{array}\right)}$

$\displaystyle \mathtt{A+3I_3=\left(\begin{array}{ccc}\mathtt{-1}&\mathtt2&\mathtt0\\\mathtt1&\mathtt0&\mathtt1\\\mathtt{-2}&\mathtt1&\mathtt{-4}\end{array}\right)+\left(\begin{array}{ccc}\mathtt{3}&\mathtt0&\mathtt0\\\mathtt0&\mathtt3&\mathtt0\\\mathtt0&\mathtt0&\mathtt3\end{array}\right)=\left(\begin{array}{ccc}\mathtt{2}&\mathtt2&\mathtt0\\\mathtt1&\mathtt3&\mathtt0\\\mathtt{-2}&\mathtt1&\mathtt{-1}\end{array}\right)}$

$\displaystyle \mathtt{2)~A=\left(\begin{array}{ccc}\mathtt{1}&\mathtt{2i}&\mathtt{-1}\\\mathtt4&\mathtt2&\mathtt{3i}\end{array}\right);~B=\left(\begin{array}{ccc}\mathtt{2i}&\mathtt4&\mathtt0\\\mathtt1&\mathtt i&\mathtt{i+1}\end{array}\right)~~~~~~~~~2iA-3iB=?}$

$\displaystyle \mathtt{2iA=2i\cdot\left(\begin{array}{ccc}\mathtt{1}&\mathtt{2i}&\mathtt{-1}\\\mathtt4&\mathtt2&\mathtt{3i}\end{array}\right)=\left(\begin{array}{ccc}\mathtt{2i}&\mathtt{-4}&\mathtt{-2i}\\\mathtt{8i}&\mathtt{4i}&\mathtt{-6}\end{array}\right)}$

$\displaystyle \mathtt{3iB=3i\cdot \left(\begin{array}{ccc}\mathtt{2i}&\mathtt4&\mathtt0\\\mathtt1&\mathtt i&\mathtt{i+1}\end{array}\right)=\left(\begin{array}{ccc}\mathtt{-6}&\mathtt{12i}&\mathtt0\\\mathtt{3i}&\mathtt{-3}&\mathtt{-3+3i}\end{array}\right)}$

$\displaystyle \mathtt{2iA-3iB=\left(\begin{array}{ccc}\mathtt{2i}&\mathtt{-4}&\mathtt{-2i}\\\mathtt{8i}&\mathtt{4i}&\mathtt{-6}\end{array}\right)-\left(\begin{array}{ccc}\mathtt{-6}&\mathtt{12i}&\mathtt0\\\mathtt{3i}&\mathtt{-3}&\mathtt{-3+3i}\end{array}\right)=}\\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{2i+6}&\mathtt{-4-12i}&\mathtt{-2i}\\\mathtt{5i}&\mathtt{4i+3}&\mathtt{-3+3i}\end{array}\right)}$

$\displaystyle \mathtt{3)~ \left(\begin{array}{ccc}\mathtt2&\mathtt1&\mathtt1\\\mathtt1&\mathtt2&\mathtt1\\\mathtt1&\mathtt1&\mathtt2\end{array}\right)\cdot \left(\begin{array}{ccc}\mathtt x\\\mathtt y\\\mathtt z\end{array}\right)=\left(\begin{array}{ccc}\mathtt4\\\mathtt1\\\mathtt3\end{array}\right)}\\ \\ \mathtt{ \left\{\begin{array}{ccc}\mathtt{2x+y+z=4}\\\mathtt{x+2y+z=1}\\\mathtt{x+y+2z=3}\end{array}\right}$

$\displaystyle \mathtt{\Delta= \left|\begin{array}{ccc}\mathtt2&\mathtt1&\mathtt1\\\mathtt1&\mathtt2&\mathtt1\\\mathtt1&\mathtt1&\mathtt2\end{array}\right|=2\cdot2\cdot2+1\cdot1\cdot1+1\cdot1\cdot1-1\cdot2\cdot1-1\cdot1\cdot2-}\\ \\ \mathtt{-2\cdot1\cdot1=8+1+1-2-2-2=10-6=4}\\ \\ \mathtt{\Delta=4 \neq 0}$

$\displaystyle \mathtt{\Delta_x= \left|\begin{array}{ccc}\mathtt4&\mathtt1&\mathtt1\\\mathtt1&\mathtt2&\mathtt1\\\mathtt3&\mathtt1&\mathtt2\end{array}\right|=4\cdot2\cdot2+1\cdot1\cdot1+1\cdot1\cdot3-1\cdot2\cdot3-1\cdot1\cdot2-}\\ \\ \mathtt{-4\cdot1\cdot1=16+1+3-6-2-4=20-12=8}\\ \\ \mathtt{\Delta_x=8}$

$\displaystyle \mathtt{\Delta_y= \left|\begin{array}{ccc}\mathtt2&\mathtt4&\mathtt1\\\mathtt1&\mathtt1&\mathtt1\\\mathtt1&\mathtt3&\mathtt2\end{array}\right|=2\cdot1\cdot2+1\cdot1\cdot3+4\cdot1\cdot1-1\cdot1\cdot1-4\cdot1\cdot2-}\\ \\ \mathtt{-2\cdot1\cdot3=4+3+4-1-8-6=11-15=-4}$

$\displaystyle \mathtt{\Delta_z= \left|\begin{array}{ccc}\mathtt2&\mathtt1&\mathtt4\\\mathtt1&\mathtt2&\mathtt1\\\mathtt1&\mathtt1&\mathtt3\end{array}\right|=2\cdot2\cdot3+4\cdot1\cdot1+1\cdot1\cdot1-4\cdot2\cdot1-1\cdot1\cdot3-}\\ \\ \mathtt{-2\cdot1\cdot1=12+4+1-8-3-2=17-13=4}$

$\displaystyle \mathtt{x= \frac{\Delta_x}{\Delta}= \frac{8}{4} =2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x=2 }~\\ \\ \mathtt{y= \frac{\Delta_y}{\Delta}= \frac{-4}{4}=-1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~y=-1} \\ \\ \mathtt{z= \frac{\Delta_z}{\Delta} = \frac{4}{4}=1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~z=1 }$