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[tex]\displaystyle\mathtt{TESTUL~1}\\\\\mathtt{1.~~A=\left(\begin{array}{ccc}\mathtt2&\mathtt3\\\mathtt3&\mathtt{-2}\end{array}\right),~B=\left(\begin{array}{ccc}\mathtt4&\mathtt{-6}\\\mathtt6&\mathtt{-4}\end{array}\right)}\\
\\ \mathtt{a)~~S=a_{11}+2a_{21}-3b_{12}+b_{22}}\\ \\
\mathtt{a_{11}=2,~a_{21}=3,~b_{12}=-6,~b_{22}=-4}\\ \\
\mathtt{S=2+2\cdot3-3\cdot(-6)+(-4)}\\ \\ \mathtt{S=2+6+18-4}\\ \\
\mathtt{S=22}\\ \\ \mathtt{b)~~b_{11}=4,~b_{22}=-4}\\ \\
\mathtt{Tr(A)=2+(-2)=2-2=0}[/tex]
[tex]\displaystyle \mathtt{c)~~xA=B,~x=?}\\ \\
\mathtt{x\left(\begin{array}{ccc}\mathtt2&\mathtt3\\\mathtt3&\mathtt{-2}\end{array}\right)=\left(\begin{array}{ccc}\mathtt4&\mathtt{-6}\\\mathtt6&\mathtt{-4}\end{array}\right)\Rightarrow
\left(\begin{array}{ccc}\mathtt{2x}&\mathtt{3x}\\\mathtt{3x}&\mathtt{-2x}\end{array}\right)=\left(\begin{array}{ccc}\mathtt4&\mathtt{-6}\\\mathtt6&\mathtt{-4}\end{array}\right)}
\\ \\ \mathtt{Nu~sunt~solu\c{t}ii}[/tex]
[tex]\displaystyle
\mathtt{2.~~2\left(\begin{array}{ccc}\mathtt3&\mathtt x\\\mathtt
x&\mathtt
y\end{array}\right)+y\left(\begin{array}{ccc}\mathtt2&\mathtt1\\\mathtt1&\mathtt0\end{array}\right)=\left(\begin{array}{ccc}\mathtt8&\mathtt3\\\mathtt3&\mathtt2\end{array}\right)}[/tex]
[tex]\displaystyle
\mathtt{\left(\begin{array}{ccc}\mathtt6&\mathtt{2x}\\\mathtt{2x}&\mathtt{2y}\end{array}\right)+\left(\begin{array}{ccc}\mathtt{2y}&\mathtt
y\\\mathtt
y&\mathtt0\end{array}\right)=\left(\begin{array}{ccc}\mathtt8&\mathtt3\\\mathtt3&\mathtt2\end{array}\right)}[/tex]
[tex]\displaystyle \mathtt{
\left\{\begin{array}{ccc}\mathtt{6+2y=8\Rightarrow2y=8-6\Rightarrow
2y=2\Rightarrow y=1}\\\mathtt{2x+y=3\Rightarrow 2x+1=3\Rightarrow
2x=3-1\Rightarrow 2x=2\Rightarrow x=1}\\\mathtt{2x+y=3\Rightarrow
2\cdot1+y=3\Rightarrow2+y=3\Rightarrow y=3-2\Rightarrow
y=1}\\\mathtt{2y=2\Rightarrow y=1}\end{array}\right}\\ \\ \\
\mathtt{x=1,~y=1}[/tex]
[tex]\displaystyle \mathtt{3.~~A=
\left(\begin{array}{ccc}\mathtt2&\mathtt{-1}&\mathtt1\\\mathtt3&\mathtt0&\mathtt1\end{array}\right),~B=\left(\begin{array}{ccc}\mathtt1&\mathtt2\\\mathtt0&\mathtt3\end{array}\right)}\\
\\ \mathtt{a)~~2\cdot BA=?~~~~~~~~~B^2=?}[/tex]
[tex]\displaystyle
\mathtt{BA=\left(\begin{array}{ccc}\mathtt1&\mathtt2\\\mathtt0&\mathtt3\end{array}\right)\left(\begin{array}{ccc}\mathtt2&\mathtt{-1}&\mathtt1\\\mathtt3&\mathtt0&\mathtt1\end{array}\right)=}\\
\\
\mathtt{=\left(\begin{array}{ccc}\mathtt{1\cdot2+2\cdot3}&\mathtt{1\cdot(-1)+2\cdot0}&\mathtt{1\cdot1+2\cdot1}\\\mathtt{0\cdot2+3\cdot3}&\mathtt{0\cdot(-1)+3\cdot0}&\mathtt{0\cdot1+3\cdot1}\end{array}\right)=}[/tex]
[tex]\displaystyle
\mathtt{=\left(\begin{array}{ccc}\mathtt{2+6}&\mathtt{-1+0}&\mathtt{1+2}\\\mathtt{0+9}&\mathtt{0+0}&\mathtt{0+3}\end{array}\right)=\left(\begin{array}{ccc}\mathtt8&\mathtt{-1}&\mathtt3\\\mathtt9&\mathtt0&\mathtt3\end{array}\right)}[/tex]
[tex]\displaystyle \mathtt{2\cdot
BA=2\left(\begin{array}{ccc}\mathtt8&\mathtt{-1}&\mathtt3\\\mathtt9&\mathtt0&\mathtt3\end{array}\right)=\left(\begin{array}{ccc}\mathtt{2\cdot8}&\mathtt{2\cdot(-1)}&\mathtt{2\cdot3}\\\mathtt{2\cdot9}&\mathtt{2\cdot0}&\mathtt{2\cdot
3}\end{array}\right)=}\\ \\
\mathtt{=\left(\begin{array}{ccc}\mathtt{16}&\mathtt{-2}&\mathtt6\\\mathtt{18}&\mathtt0&\mathtt6\end{array}\right)}[/tex]
[tex]\displaystyle
\mathtt{=\left(\begin{array}{ccc}\mathtt{1+0}&\mathtt{2+6}\\\mathtt{0+0}&\mathtt{0+9}\end{array}\right)=\left(\begin{array}{ccc}\mathtt1&\mathtt8\\\mathtt0&\mathtt9\end{array}\right)}[/tex]
[tex]\displaystyle \mathtt{b)~~^tA\cdot B=?}\\ \\
\mathtt{A=\left(\begin{array}{ccc}\mathtt2&\mathtt{-1}&\mathtt1\\\mathtt3&\mathtt0&\mathtt1\end{array}\right)\Rightarrow
^tA=\left(\begin{array}{ccc}\mathtt2&\mathtt3\\\mathtt{-1}&\mathtt0\\\mathtt1&\mathtt1\end{array}\right)}[/tex]
[tex]\displaystyle \mathtt{^tA\cdot
B=\left(\begin{array}{ccc}\mathtt2&\mathtt3\\\mathtt{-1}&\mathtt0\\\mathtt1&\mathtt1\end{array}\right)\left(\begin{array}{ccc}\mathtt1&\mathtt2\\\mathtt0&\mathtt3\end{array}\right)=\left(\begin{array}{ccc}\mathtt{2\cdot1+3\cdot0}&\mathtt{2\cdot2+3\cdot3}\\\mathtt{-1\cdot1+0\cdot0}&\mathtt{-1\cdot2+0\cdot3}\\\mathtt{1\cdot1+1\cdot0}&\mathtt{1\cdot2+1\cdot3}\end{array}\right)=}[/tex]
[tex]\displaystyle
\mathtt{=\left(\begin{array}{ccc}\mathtt{2+0}&\mathtt{4+9}\\\mathtt{-1+0}&\mathtt{-2+0}\\\mathtt{1+0}&\mathtt{2+3}\end{array}\right)=\left(\begin{array}{ccc}\mathtt2&\mathtt{13}\\\mathtt{-1}&\mathtt{-2}\\\mathtt1&\mathtt5\end{array}\right)}[/tex]
