Question

Testul 1 , exercitiul 2 !! Va rogg urgentt..
Este vorba despre matrice , clasa a 12 a.
Am nevoie urgent.. Dau coroana...

Answer 1

$\displaystyle \mathtt{Testul~1}\\ \\ \mathtt{2.~~~A=\left(\begin{array}{ccc}\mathtt3&\mathtt4\\\mathtt2&\mathtt3\\\end{array}\right);~B=\left(\begin{array}{ccc}\mathtt3&\mathtt{-4}\\\mathtt{-2}&\mathtt3\\\end{array}\right)}\\ \\ \mathtt{a)A+B=6I_2}$

$\displaystyle \mathtt{A+B=\left(\begin{array}{ccc}\mathtt3&\mathtt4\\\mathtt2&\mathtt3\\\end{array}\right)+\left(\begin{array}{ccc}\mathtt3&\mathtt{-4}\\\mathtt{-2}&\mathtt3\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{3+3}&\mathtt{4+(-4)}\\\mathtt{2+(-2)}&\mathtt{3+3}\\\end{array}\right)=}\\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt6&\mathtt0\\\mathtt0&\mathtt6\\\end{array}\right)}$

$\displaystyle \mathtt{A+B=\left(\begin{array}{ccc}\mathtt6&\mathtt0\\\mathtt0&\mathtt6\\\end{array}\right)=6 \cdot \left(\begin{array}{ccc}\mathtt1&\mathtt0\\\mathtt0&\mathtt1\\\end{array}\right)=6I_2}$

$\displaystyle \mathtt{b)A^3+B^2=?;~A^2+B^2=34I_2}\\ \\ \mathtt{A^3=A^2\cdot A}\\ \\ \mathtt{A^2=A \cdot A=\left(\begin{array}{ccc}\mathtt3&\mathtt4\\\mathtt2&\mathtt3\\\end{array}\right) \cdot \left(\begin{array}{ccc}\mathtt3&\mathtt4\\\mathtt2&\mathtt3\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{3\cdot3+4\cdot2}&\mathtt{3 \cdot4+4\cdot3}\\\mathtt{2\cdot3+3\cdot2}&\mathtt{2\cdot4+3\cdot3}\\\end{array}\right)=}$

$\displaystyle \mathtt{=\left(\begin{array}{ccc}\mathtt{9+8}&\mathtt{12+12}\\\mathtt{6+6}&\mathtt{8+9}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{17}&\mathtt{24}\\\mathtt{12}&\mathtt{17}\\\end{array}\right)}$

$\displaystyle \mathtt{A^3=A^2\cdot A=\left(\begin{array}{ccc}\mathtt{17}&\mathtt{24}\\\mathtt{12}&\mathtt{17}\\\end{array}\right)\cdot \left(\begin{array}{ccc}\mathtt3&\mathtt4\\\mathtt2&\mathtt3\\\end{array}\right)=}\\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{17 \cdot 3+24\cdot2}&\mathtt{17\cdot4+24\cdot3}\\\mathtt{12\cdot3+17\cdot2}&\mathtt{12\cdot4+17\cdot3}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{51+48}&\mathtt{68+72}\\\mathtt{36+34}&\mathtt{48+51}\\\end{array}\right)=}$

$\displaystyle \mathtt{=\left(\begin{array}{ccc}\mathtt{99}&\mathtt{140}\\\mathtt{70}&\mathtt{99}\\\end{array}\right)}$

$\displaystyle \mathtt{B^2=B \cdot B=\left(\begin{array}{ccc}\mathtt3&\mathtt{-4}\\\mathtt{-2}&\mathtt3\\\end{array}\right)\cdot \left(\begin{array}{ccc}\mathtt3&\mathtt{-4}\\\mathtt{-2}&\mathtt3\\\end{array}\right)=}\\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{3 \cdot 3+(-4)\cdot(-2)}&\mathtt{3\cdot(-4)+(-4)\cdot3}\\\mathtt{(-2)\cdot3+3\cdot(-2)}&\mathtt{(-2)\cdot(-4)+3\cdot3}\\\end{array}\right)=}$

$\displaystyle \mathtt{=\left(\begin{array}{ccc}\mathtt{9+8}&\mathtt{(-12)-12}\\\mathtt{(-6)-6}&\mathtt{8+9}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{17}&\mathtt{-24}\\\mathtt{-12}&\mathtt{17}\\\end{array}\right)}$

$\displaystyle \mathtt{A^3+B^2=\left(\begin{array}{ccc}\mathtt{99}&\mathtt{140}\\\mathtt{70}&\mathtt{99}\\\end{array}\right)+\left(\begin{array}{ccc}\mathtt{17}&\mathtt{-24}\\\mathtt{-12}&\mathtt{17}\\\end{array}\right)=}\\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{99+17}&\mathtt{140+(-24)}\\\mathtt{70+(-12)}&\mathtt{99+17}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{116}&\mathtt{116}\\\mathtt{58}&\mathtt{116}\\\end{array}\right)}$

$\displaystyle \mathtt{A^3+B^2=\left(\begin{array}{ccc}\mathtt{116}&\mathtt{116}\\\mathtt{58}&\mathtt{116}\\\end{array}\right)}$

$\displaystyle \mathtt{A^2+B^2=\left(\begin{array}{ccc}\mathtt{17}&\mathtt{24}\\\mathtt{12}&\mathtt{17}\\\end{array}\right)+\left(\begin{array}{ccc}\mathtt{17}&\mathtt{-24}\\\mathtt{-12}&\mathtt{17}\\\end{array}\right)=}\\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{17+17}&\mathtt{24+(-24)}\\\mathtt{12+(-12)}&\mathtt{17+17}\\\end{array}\right)=\left(\begin{array}{ccc}\mathtt{34}&\mathtt{0}\\\mathtt{0}&\mathtt{34}\\\end{array}\right)}$

$\displaystyle \mathtt{A^2+B^2=\left(\begin{array}{ccc}\mathtt{34}&\mathtt{0}\\\mathtt{0}&\mathtt{34}\\\end{array}\right)=34 \cdot\left(\begin{array}{ccc}\mathtt{1}&\mathtt{0}\\\mathtt{0}&\mathtt{1}\\\end{array}\right)=34I_2}$