Question

Multiplicação de matrizes dúvida..................................

Answer 1

Explicação passo-a-passo:

MULTIPLICAÇÃO DE MATRIZES

  * Matriz por matriz

    Sejam as matrizes A, B e C (matriz resultante da multiplicação)

   

         $A=\left[\begin{array}{ccc}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{array}\right]$    ,   $B=\left[\begin{array}{ccc}b_{11}&b_{12}\\b_{21}&b_{22}\\\end{array}\right]$    ,   $C=\left[\begin{array}{ccc}c_{11}&c_{12}\\c_{21}&c_{22}\\\end{array}\right]$  

   

    A multiplicação é feita assim

         $c_{11}=a_{11}.b_{11}+a_{12}.b_{21}$

         $c_{12}=a_{11}.b_{12}+a_{12}.b_{22}$

         $c_{21}=a_{21}.b_{11}+a_{22}.b_{21}$

         $c_{22}=a_{21}.b_{12}+a_{22}.b_{22}$

  * Número por matriz

    Seja o número x e as matrizes A e B (matriz resultante da multiplicação)

         $A=\left[\begin{array}{ccc}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{array}\right]$    ,   $B=\left[\begin{array}{ccc}b_{11}&b_{12}\\b_{21}&b_{22}\\\end{array}\right]$

         $x.\left[\begin{array}{ccc}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{array}\right]=\left[\begin{array}{ccc}b_{11} &b_{12}\\b_{21}&b_{22}\\\end{array}\right]$

    Multiplique o número x com cada elemento da matriz

         $b_{11}=x.a_{11}$

         $b_{12}=x.a_{12}$

         $b_{21}=x.a_{21}$

         $b_{22}=x.a_{22}$

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1)

   $A=\left[\begin{array}{ccc}-3&0\\1&-2\\\end{array}\right]$    ,   $B=\left[\begin{array}{ccc}1&-4\\2&0\\\end{array}\right]$    ,   $C=\left[\begin{array}{ccc}10&30\\-20&25\\\end{array}\right]$

   $A.B-\frac{1}{5}.C$

   $\left[\begin{array}{ccc}-3&0\\1&-2\\\end{array}\right].\left[\begin{array}{ccc}1&-4\\2&0\\\end{array}\right]-\frac{1}{5}.\left[\begin{array}{ccc}10&30\\-20&25\\\end{array}\right]$

   $\left[\begin{array}{ccc}-3.1+0.2&-3.(-4)+0.0\\1.1+(-2).2&1.(-4)+(-2).0\\\end{array}\right]-\left[\begin{array}{ccc}\frac{1}{5}.10 &\frac{1}{5}.30 \\\frac{1}{5}.(-20)&\frac{1}{5}.25\\\end{array}\right]$

   $\left[\begin{array}{ccc}-3+0&12+0\\1-4&-4-0\\\end{array}\right]-\left[\begin{array}{ccc}2&6\\-4&5\\\end{array}\right]$

   $\left[\begin{array}{ccc}-3&12\\-3&-4\\\end{array}\right]-\left[\begin{array}{ccc}2&6\\-4&5\\\end{array}\right]$

   $\left[\begin{array}{ccc}-3-2&12-6\\-3-(-4)&-4-5\\\end{array}\right]=\left[\begin{array}{ccc}-5&6\\1&-9\\\end{array}\right]$

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2)

   $A=\left[\begin{array}{ccc}1&-1\\2&3\\\end{array}\right]$    ,   $B=\left[\begin{array}{ccc}1&0\\-1&2\\\end{array}\right]$    ,   $C=\left[\begin{array}{ccc}1&3\\0&-1\\\end{array}\right]$

   $A.B-3.C$

   $\left[\begin{array}{ccc}1&-1\\2&3\\\end{array}\right].\left[\begin{array}{ccc}1&0\\-1&2\\\end{array}\right]-3.\left[\begin{array}{ccc}1&3\\0&-1\\\end{array}\right]$

   $\left[\begin{array}{ccc}1.1+(-1).(-1)&1.0+(-1).2\\2.1+3.(-1)&2.0+3.2\\\end{array}\right]-\left[\begin{array}{ccc}3.1&3.3\\3.0&3.(-1)\\\end{array}\right]$

   $\left[\begin{array}{ccc}1+1&0-2\\2-3&0+6\\\end{array}\right]-\left[\begin{array}{ccc}3&9\\0&-3\\\end{array}\right]$

   $\left[\begin{array}{ccc}2&-2\\-1&6\\\end{array}\right]-\left[\begin{array}{ccc}3&9\\0&-3\\\end{array}\right]$

   $\left[\begin{array}{ccc}2-3&-2-9\\-1-0&6-(-3)\\\end{array}\right]=\left[\begin{array}{ccc}-1&-11\\-1&9\\\end{array}\right]$

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c)

   $A=\left[\begin{array}{ccc}-2&4\\10&-8\\\end{array}\right]$    ,   $B=\left[\begin{array}{ccc}-1&5\\3&2\\\end{array}\right]$    ,   $C=\left[\begin{array}{ccc}1&2\\0&-1\\\end{array}\right]$

   $B.C-\frac{1}{2}.A$

   $\left[\begin{array}{ccc}-1&5\\3&2\\\end{array}\right].\left[\begin{array}{ccc}1&2\\0&-1\\\end{array}\right]-\frac{1}{2}.\left[\begin{array}{ccc}-2&4\\10&-8\\\end{array}\right]$

   $\left[\begin{array}{ccc}-1.1+5.0&-1.2+5.(-1)\\3.1+2.0&3.2+2.(-1)\\\end{array}\right]-\left[\begin{array}{ccc}\frac{1}{2}.(-2)&\frac{1}{2}.4\\\frac{1}{2}.10&\frac{1}{2}.(-8) \\\end{array}\right]$

   $\left[\begin{array}{ccc}-1+0&-2-5\\3+0&6-2\\\end{array}\right]-\left[\begin{array}{ccc}-1&2\\5&-4\\\end{array}\right]$

   $\left[\begin{array}{ccc}-1&-7\\3&4\\\end{array}\right]-\left[\begin{array}{ccc}-1&2\\5&-4\\\end{array}\right]$

   $\left[\begin{array}{ccc}-1-(-1)&-7-2\\3-5&4-(-4)\\\end{array}\right]=\left[\begin{array}{ccc}0&-9\\-2&8\\\end{array}\right]$