Question

Testul 1 , exercitiul 1 !! Vaa rooog urgenttttttt !!!
Dau coroanaaaa..
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Answer 1

[tex]\displaystyle \mathtt{Testul~1}\\ \\ \mathtt{1.~~~A=\left(\begin{array}{ccc}\mathtt1&\mathtt2\\\mathtt{-3}&\mathtt5\\\end{array}\right);~B=\left(\begin{array}{ccc}\mathtt4&\mathtt6\\\mathtt8&\mathtt3\\\end{array}\right);~I_2=\left(\begin{array}{ccc}\mathtt1&\mathtt0\\\mathtt0&\mathtt1\\\end{array}\right)}[/tex]

[tex]\displaystyle \mathtt{2A-3B=?}\\ \\ \mathtt{2A=2\cdot\left(\begin{array}{ccc}\mathtt1&\mathtt2\\\mathtt{-3}&\mathtt5\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{2\cdot1}&\mathtt{2\cdot2}\\\mathtt{2\cdot(-3)}&\mathtt{2\cdot5}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt2&\mathtt4\\\mathtt{-6}&\mathtt{10}\\\end{array}\right)}[/tex]

[tex]\displaystyle \mathtt{3B=3 \cdot \left(\begin{array}{ccc}\mathtt4&\mathtt6\\\mathtt8&\mathtt3\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{3\cdot4}&\mathtt{3\cdot6}\\\mathtt{3\cdot8}&\mathtt{3\cdot3}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{12}&\mathtt{18}\\\mathtt{24}&\mathtt{9}\\\end{array}\right)}[/tex]

[tex]\displaystyle \mathtt{2A-3B=\left(\begin{array}{ccc}\mathtt2&\mathtt4\\\mathtt{-6}&\mathtt{10}\\\end{array}\right)-\left(\begin{array}{ccc}\mathtt{12}&\mathtt{18}\\\mathtt{24}&\mathtt{9}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{2-12}&\mathtt{4-18}\\\mathtt{(-6)-24}&\mathtt{10-9}\\\end{array}\right)=}\\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{-10}&\mathtt{-14}\\\mathtt{-30}&\mathtt1\\\end{array}\right)}[/tex]

[tex]\displaystyle \mathtt{2A-3B=\left(\begin{array}{ccc}\mathtt{-10}&\mathtt{-14}\\\mathtt{-30}&\mathtt1\\\end{array}\right) }[/tex]

[tex]\displaystyle \mathtt{b)A^2-B+I_2=?;~A^3=?}\\\\\mathtt{A^2=A\cdot A=\left(\begin{array}{ccc}\mathtt1&\mathtt2\\\mathtt{-3}&\mathtt5\\\end{array}\right)\cdot\left(\begin{array}{ccc}\mathtt1&\mathtt2\\\mathtt{-3}&\mathtt5\\\end{array}\right)=}\\\\\mathtt{=\left(\begin{array}{ccc}\mathtt{1\cdot1+2\cdot(-3)}&\mathtt{1\cdot2+2\cdot5}\\\mathtt{(-3)\cdot1+5\cdot(-3)}&\mathtt{(-3)\cdot2+5\cdot5}\\\end{array}\right)=}[/tex]

[tex]\displaystyle \mathtt{=\left(\begin{array}{ccc}\mathtt{1-6}&\mathtt{2+10}\\\mathtt{(-3)-15}&\mathtt{(-6)+25}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{-5}&\mathtt{12}\\\mathtt{-18}&\mathtt{19}\\\end{array}\right)}[/tex]

$\displaystyle \mathtt{A^2-B=\left(\begin{array}{ccc}\mathtt{-5}&\mathtt{12}\\\mathtt{-18}&\mathtt{19}\\\end{array}\right)-\left(\begin{array}{ccc}\mathtt4&\mathtt6\\\mathtt8&\mathtt3\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{(-5)-4}&\mathtt{12-6}\\\mathtt{(-18)-8}&\mathtt{19-3}\\\end{array}\right)=}\\ \\ =\mathtt{\left(\begin{array}{ccc}\mathtt{-9}&\mathtt6\\\mathtt{-26}&\mathtt{16}\\\end{array}\right)}$

$\displaystyle \mathtt{A^2-B+I_2= \left(\begin{array}{ccc}\mathtt{-9}&\mathtt6\\\mathtt{-26}&\mathtt{16}\\\end{array}\right)+\left(\begin{array}{ccc}\mathtt1&\mathtt0\\\mathtt0&\mathtt1\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{(-9)+1}&\mathtt{6+0}\\\mathtt{(-26)+0}&\mathtt{16+1}\\\end{array}\right)=} \\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{-8}&\mathtt6\\\mathtt{-26}&\mathtt{17}\\\end{array}\right)}$

$\displaystyle \mathtt{A^2-B+I_2=\left(\begin{array}{ccc}\mathtt{-8}&\mathtt6\\\mathtt{-26}&\mathtt{17}\\\end{array}\right) }$

$\displaystyle \mathtt{A^3=A^2\cdot A=\left(\begin{array}{ccc}\mathtt{-5}&\mathtt{12}\\\mathtt{-18}&\mathtt{19}\\\end{array}\right)\cdot \left(\begin{array}{ccc}\mathtt1&\mathtt2\\\mathtt{-3}&\mathtt5\\\end{array}\right)=}$

$\displaystyle \mathtt{=\left(\begin{array}{ccc}\mathtt{(-5) \cdot1+12\cdot(-3)}&\mathtt{(-5)\cdot2+12\cdot5}\\\mathtt{(-18)\cdot1+19\cdot(-3)}&\mathtt{(-18)\cdot2+19\cdot5}\\\end{array}\right)=} \\ \\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{(-5)-36}&\mathtt{(-10)+60}\\\mathtt{(-18)-57}&\mathtt{(-36)+95}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{-41}&\mathtt{50}\\\mathtt{-75}&\mathtt{59}\\\end{array}\right)}$

$\displaystyle \mathtt{A^3=\left(\begin{array}{ccc}\mathtt{-41}&\mathtt{50}\\\mathtt{-75}&\mathtt{59}\\\end{array}\right)}$