Question

Inversa matricei (A-B)^2

Answer 1

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Answer 2

$\displaystyle \mathtt{A= \left(\begin{array}{ccc}\mathtt2&\mathtt{-1}&\mathtt{-1}\\\mathtt{-1}&\mathtt2&\mathtt{-1}\\\mathtt{-1}&\mathtt{-1}&\mathtt2\end{array}\right);~B=\left(\begin{array}{ccc}\mathtt{-1}&\mathtt{-1}&\mathtt{-1}\\\mathtt{-1}&\mathtt{-1}&\mathtt{-1}\\\mathtt{-1}&\mathtt{-1}&\mathtt{-1}\end{array}\right)}$

$\displaystyle \mathtt{A-B=\left(\begin{array}{ccc}\mathtt2&\mathtt{-1}&\mathtt{-1}\\\mathtt{-1}&\mathtt2&\mathtt{-1}\\\mathtt{-1}&\mathtt{-1}&\mathtt2\end{array}\right)-\left(\begin{array}{ccc}\mathtt{-1}&\mathtt{-1}&\mathtt{-1}\\\mathtt{-1}&\mathtt{-1}&\mathtt{-1}\\\mathtt{-1}&\mathtt{-1}&\mathtt{-1}\end{array}\right)=}$

$\displaystyle \mathtt{=\left(\begin{array}{ccc}\mathtt{2-(-1)}&\mathtt{-1-(-1)}&\mathtt{-1-(-1)}\\\mathtt{-1-(-1)}&\mathtt{2-(-1)}&\mathtt{-1-(-1)}\\\mathtt{-1-(-1)}&\mathtt{-1-(-1)&\mathtt{2-(-1)}\end{array}\right )}= \left(\begin{array}{ccc}\mathtt3&\mathtt0&\mathtt0\\\mathtt0&\mathtt3&\mathtt0\\\mathtt0&\mathtt0&\mathtt3\end{array}\right)}$

$\displaystyle\mathtt{A-B=\left(\begin{array}{ccc}\mathtt3&\mathtt0&\mathtt0\\\mathtt0&\mathtt3&\mathtt0\\\mathtt0&\mathtt0&\mathtt3\end{array}\right)}\\\\\mathtt{(A-B)^2=\left(\begin{array}{ccc}\mathtt3&\mathtt0&\mathtt0\\\mathtt0&\mathtt3&\mathtt0\\\mathtt0&\mathtt0&\mathtt3\end{array}\right)\cdot\left(\begin{array}{ccc}\mathtt3&\mathtt0&\mathtt0\\\mathtt0&\mathtt3&\mathtt0\\\mathtt0&\mathtt0&\mathtt3\end{array}\right)=}$

$\displaystyle\mathtt{=\left(\begin{array}{ccc}\mathtt{3\cdot3+0\cdot0+0\cdot 0}&\mathtt{3 \cdot0+0\cdot3+0\cdot 0}&\mathtt{3\cdot0+0\cdot0+0\cdot 3}\\\mathtt{0\cdot 3+3 \cdot0+0\cdot 0}&\mathtt{0\cdot0+3\cdot3+0\cdot0}&\mathtt{0\cdot0+3\cdot0+0\cdot3}\\\mathtt{0\cdot3+0\cdot0+3\cdot0}&\mathtt{0\cdot0+0\cdot3+3\cdot0}&\mathtt{0\cdot 0+0\cdot0+3\cdot3}\end{array}\right )=}$

$\displaystyle \mathtt{=\left(\begin{array}{ccc}\mathtt{9+0+0}&\mathtt{0+0+0}&\mathtt{0+0+0}\\\mathtt{0+0+0}&\mathtt{0+9+0}&\mathtt{0+0+0}\\\mathtt{0+0+0}&\mathtt{0+0+0}&\mathtt{0+0+9}\end{array}\right)=\left(\begin{array}{ccc}\mathtt9&\mathtt0&\mathtt0\\\mathtt0&\mathtt9&\mathtt0\\\mathtt0&\mathtt0&\mathtt9\end{array}\right)}$

$\displaystyle \mathtt{(A-B)^2=\left(\begin{array}{ccc}\mathtt9&\mathtt0&\mathtt0\\\mathtt0&\mathtt9&\mathtt0\\\mathtt0&\mathtt0&\mathtt9\end{array}\right)}\\\\\mathtt{det((A-B)^2)=\left|\begin{array}{ccc}\mathtt9&\mathtt0&\mathtt0\\\mathtt0&\mathtt9&\mathtt0\\\mathtt0&\mathtt0&\mathtt9\end{array}\right|=9\cdot9\cdot9+0 \cdot0\cdot0+0\cdot0\cdot0-0\cdot9\cdot 0-}\\ \\ \mathtt{-0 \cdot 0\cdot 9-9 \cdot 0\cdot0=729}\\\\\mathtt{det((A-B)^2)=729}$

$\displaystyle\mathtt{((A-B)^2)^T=\left(\begin{array}{ccc}\mathtt9&\mathtt0&\mathtt0\\\mathtt0&\mathtt9&\mathtt0\\\mathtt0&\mathtt0&\mathtt9\end{array}\right)}$

$\displaystyle \mathtt{D_{11}=(-1)^{1+1} \cdot \left|\begin{array}{ccc}\mathtt9&\mathtt0\\\mathtt0&\mathtt9\end{array}\right|=1 \cdot 81=81}\\ \\ \mathtt{D_{12}=(-1)^{1+2} \cdot \left|\begin{array}{ccc}\mathtt0&\mathtt0\\\mathtt0&\mathtt9\end{array}\right|=(-1)\cdot0=0}\\ \\ \mathtt{D_{13}=(-1)^{1+3}\cdot \left|\begin{array}{ccc}\mathtt0&\mathtt9\\\mathtt0&\mathtt0\end{array}\right|=1 \cdot 0=0}$
[tex]\displaystyle \mathtt{D_{21}=(-1)^{2+1}\cdot \left|\begin{array}{ccc}\mathtt0&\mathtt0\\\mathtt0&\mathtt9\end{array}\right|=(-1)\cdot0=0}\\ \\ \mathtt{D_{22}=(-1)^{2+2}\cdot \left|\begin{array}{ccc}\mathtt9&\mathtt0\\\mathtt0&\mathtt9\end{array}\right|=1\cdot 81=81}\\\\ \mathtt{D_{23}=(-1)^{2+3}\cdot \left|\begin{array}{ccc}\mathtt9&\mathtt0\\\mathtt0&\mathtt0\end{array}\right|=(-1)\cdot 0=0}[/tex]
$\displaystyle \mathtt{D_{31}=(-1)^{3+1}\cdot\left|\begin{array}{ccc}\mathtt0&\mathtt0\\\mathtt9&\mathtt0\end{array}\right|=1 \cdot 0=0}\\ \\ \mathtt{D_{32}=(-1)^{3+2}\cdot \left|\begin{array}{ccc}\mathtt9&\mathtt0\\\mathtt0&\mathtt0\end{array}\right|=(-1) \cdot 0=0}\\ \\ \mathtt{D_{33}=(-1)^{3+3}\cdot \left|\begin{array}{ccc}\mathtt9&\mathtt0\\\mathtt0&\mathtt9\end{array}\right|=1 \cdot 81=81}$

$\displaystyle \mathtt{((A-B)^2)^*=\left(\begin{array}{ccc}\mathtt{81}&\mathtt0&\mathtt0\\\mathtt0&\mathtt{81}&\mathtt0\\\mathtt0&\mathtt0&\mathtt{81}\end{array}\right)}$

$\displaystyle \mathtt{((A-B)^2)^{-1}=\frac{1}{729}\cdot \left(\begin{array}{ccc}\mathtt{81}&\mathtt0&\mathtt0\\\mathtt0&\mathtt{81}&\mathtt0\\\mathtt0&\mathtt0&\mathtt{81}\end{array}\right)=\left(\begin{array}{ccc}\mathtt{ \frac{81}{729} }&\mathtt0&\mathtt0\\\mathtt0&\mathtt{ \frac{81}{729} }&\mathtt0\\\mathtt0&\mathtt0&\mathtt{ \frac{81}{729} }\end{array}\right)=}}$

$\displaystyle \mathtt{=\left(\begin{array}{ccc}\mathtt{ \frac{1}{9} }&\mathtt0&\mathtt0\\\mathtt0&\mathtt{\frac{1}{9} }&\mathtt0\\\mathtt0&\mathtt0&\mathtt{ \frac{1}{9} }\end{array}\right)}$

$\displaystyle \mathtt{((A-B)^2)^{-1}=\left(\begin{array}{ccc}\mathtt{\frac{1}{9} }&\mathtt0&\mathtt0\\\mathtt0&\mathtt{ \frac{1}{9} }&\mathtt0\\\mathtt0&\mathtt0&\mathtt{\frac{1}{9} }\end{array}\right)}$