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ȘtefanPopa
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Pe: 2020-10-29T22:27:08+02:00 In: In:

Ecuații matriceale. Cine mă poate ajuta?

Ecuații matriceale. Cine mă poate ajuta?

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    Ecuatiile sunt de forma AX=B sau XA=B. Acestea se rezolva inmultind la stanga, respectiv la dreapta, cu inversa lui A, atunci cand aceasta exista.

    1.\begin{pmatrix} 1&0\\ 0&2 \end{pmatrix}\cdot X=\begin{pmatrix}3&4 \\ -1&5 \end{pmatrix}

    Calculam inversa primei matrice, pe  care o notam cu A:

    Calculam determinantul:detA=1*2-0*0=2.

    Scriem transpusa:A^{T}=\begin{pmatrix} 1&0\\2&0 \end{pmatrix}

    Aflam adjuncta:A^*=\begin{pmatrix} (-1)^{1+1}\cdot2 & (-1)^{1+2}\cdot0 \\ (-1)^{1+2}\cdot0 & (-1)^{2+2}\cdot1 \end{pmatrix}=\begin{pmatrix}2&0\\0&1 \end{pmatrix}

    Si in final, invers:A^{-1}=\frac{1}{detA}A^*=\frac12\begin{pmatrix} 2&0\\0&1 \end{pmatrix}=\begin{pmatrix}1&0\\0&\frac12\end{pmatrix}

    Inmultim ecuatia data la stanga cu A^{-1}, acest lucru eliminand matricea A si lasand doar necunoscuta X:

    X=\begin{pmatrix}1&0\\0&\frac12\end{pmatrix}\cdot\begin{pmatrix}3&4\\-1&5\end{pmatrix}=\begin{pmatrix}1*3+0*(-1)&1*4+0*5\\0*3+\frac12*(-1)&0*4+\frac12*5\end{pmatrix}=\begin{pmatrix}3&4\\-\frac12&\frac52\end{pmatrix}

    Pentru celelalte subpuncte nu voi intra in prea multe detalii. Daca nu intelegi ceva, intreaba.

    2.X=\begin{pmatrix}1&-1\\0&1\end{pmatrix}^{-1}\cdot\begin{pmatrix}2&-3\\3&1\end{pmatrix}=\begin{pmatrix}1&1\\0&1\end{pmatrix}\cdot\begin{pmatrix}2&-3\\3&1\end{pmatrix}=\begin{pmatrix}5&-2\\3&1\end{pmatrix}

    3.X=\begin{pmatrix}0&1\\-1&2\end{pmatrix}^{-1}\cdot\begin{pmatrix}1&-1\\3&2\end{pmatrix}=\begin{pmatrix}2&-1\\1&0\end{pmatrix}\cdot\begin{pmatrix}1&-1\\3&2\end{pmatrix}=\begin{pmatrix}-1&-4\\1&-1\end{pmatrix}

    4.X=\begin{pmatrix}0&-1\\2&1\end{pmatrix}^{-1}\cdot\begin{pmatrix}3&-1&4\\1&2&5\end{pmatrix}=\begin{pmatrix}\frac12&\frac12\\-1&0\end{pmatrix}\cdot\begin{pmatrix}3&-1&4\\1&2&5\end{pmatrix}=\begin{pmatrix}2&\frac12&\frac92\\-3&1&-4\end{pmatrix}

    5.X=\begin{pmatrix}1&0\\0&-4\end{pmatrix}^{-1}\cdot\begin{pmatrix}1&0&3&-1\\5&1&2&4\end{pmatrix}=\begin{pmatrix}1&0\\0&-\frac14\end{pmatrix}\cdot\begin{pmatrix}1&0&3&-1\\5&1&2&4\end{pmatrix}=\begin{pmatrix}1&0&3&-1\\-\frac54&-\frac14&-\frac12&-1\end{pmatrix}

    6.X=\begin{pmatrix}2&-1\\0&3\end{pmatrix}^{-1}\cdot\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}\frac12&\frac16\\0&\frac13\end{pmatrix}\cdot\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}\frac43\\-\frac13\end{pmatrix}

    7.X=\begin{pmatrix}1&1\\5&0\\-1&1\end{pmatrix}\cdot\begin{pmatrix}1&-1\\0&1\end{pmatrix}^{-1}=\begin{pmatrix}1&1\\5&0\\-1&1\end{pmatrix}\cdot\begin{pmatrix}1&1\\0&1\end{pmatrix}=\begin{pmatrix}1&2\\5&5\\-1&0\end{pmatrix}

    8.X=\begin{pmatrix}1&-1\end{pmatrix}\cdot\begin{pmatrix}1&0\\0&-2\end{pmatrix}^{-1}=\begin{pmatrix}1&-1\end{pmatrix}\cdot\begin{pmatrix}1&0\\0&-\frac12\end{pmatrix}=\begin{pmatrix}1&\frac12\end{pmatrix}

    9.X=\begin{pmatrix}2&5&0\\0&1&-1\end{pmatrix}\cdot\begin{pmatrix}1&0&0\\0&2&0\\1&0&-1\end{pmatrix}^{-1}=\begin{pmatrix}2&5&0\\0&1&-1\end{pmatrix}\cdot\begin{pmatrix}1&0&0\\0&\frac12&0\\1&0&-1\end{pmatrix}=\begin{pmatrix}2&\frac52&0\\-1&\frac12&1\end{pmatrix}

    10.X=\begin{pmatrix}1&0&1\\0&-1&2\\0&0&3\end{pmatrix}^{-1}\cdot\begin{pmatrix}5&-4&2\\1&0&3\\0&1&4\end{pmatrix}=\begin{pmatrix}1&0&-\frac12\\0&-1&\frac23\\0&0&\frac13\end{pmatrix}\cdot\begin{pmatrix}5&-4&2\\1&0&3\\0&1&4\end{pmatrix}=\begin{pmatrix}5&-\frac{13}3&\frac23\\-1&\frac23&-\frac13\\0&\frac13&\frac43\end{pmatrix}

    11.X=\begin{pmatrix}1&0&-1\\2&1&0\\0&0&3\end{pmatrix}^{-1}\cdot\begin{pmatrix}2&-1\\0&5\\4&3\end{pmatrix}=\begin{pmatrix}1&0&\frac13\\-2&1&-\frac23\\0&0&\frac13\end{pmatrix}\cdot\begin{pmatrix}2&-1\\0&5\\4&3\end{pmatrix}=\begin{pmatrix}\frac{10}3&0\\-\frac{20}3&5\\\frac43&1\end{pmatrix}

    12.X=\begin{pmatrix}1&0&2\\-1&1&0\\0&0&-2\end{pmatrix}^{-1}\cdot\begin{pmatrix}2\\5\\-1\end{pmatrix}=\begin{pmatrix}1&0&1\\1&1&1\\0&0&-\frac12\end{pmatrix}\cdot\begin{pmatrix}2\\5\\-1\end{pmatrix}=\begin{pmatrix}1\\6\\\frac12\end{pmatrix}

    13.X=\begin{pmatrix}5&0&4\\-1&2&1\end{pmatrix}\cdot\begin{pmatrix}1&0&0\\0&-1&2\\2&0&3\end{pmatrix}^{-1}=\begin{pmatrix}5&0&4\\-1&2&1\end{pmatrix}\cdot\begin{pmatrix}1&0&0\\-\frac43&-1&\frac23\\-\frac23&0&\frac13\end{pmatrix}=\begin{pmatrix}\frac73&0&\frac43\\-\frac{13}3&-2&\frac53\end{pmatrix}

    14.X=\begin{pmatrix}2&0&5\end{pmatrix}\cdot\begin{pmatrix}2&0&0\\0&1&-1\\1&0&-1\end{pmatrix}^{-1}=\begin{pmatrix}2&0&5\end{pmatrix}\cdot\begin{pmatrix}\frac12&0&0\\\frac12&1&-1\\\frac12&0&-1\end{pmatrix}=\begin{pmatrix}\frac72&0&-5\end{pmatrix}

    Pentru calcularile de inverse si inmultirile de matrice de la punctele 2-14, am folosit https://matrixcalc.org/en, unde poti introduce si tu matricele si sa vezi pasii de calcul mai detaliat.

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