Question

3- Dadas as
matrizes A- (2 3/ 5 1) e B = {3 1/ 2 1) determinam
a) A²
b) B²
c) (A+B) x (A-B)​

Answer 1

Explicação passo-a-passo:

Sejam as matrizes

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

A multiplicação é feita assim

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

E a soma/subtração é feita assim

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

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

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$A=\left[\begin{array}{ccc}2&3\\5&1\\\end{array}\right]$     ;     $B=\left[\begin{array}{ccc}3&1\\2&1\\\end{array}\right]$

a) A²

   A² = A × A

   $AxA=\left[\begin{array}{ccc}2&3\\5&1\\\end{array}\right].\left[\begin{array}{ccc}2&3\\5&1\\\end{array}\right]$

   $AxA=\left[\begin{array}{ccc}2.2+3.5&2.3+3.1\\5.2+1.5&5.3+1.1\\\end{array}\right]$

   $AxA=\left[\begin{array}{ccc}4+15&6+3\\10+5&15+1\\\end{array}\right]$

   $AxA=\left[\begin{array}{ccc}19&9\\15&16\\\end{array}\right]$

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b) B²

   B² = B × B

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

   $BxB=\left[\begin{array}{ccc}3.3+1.2&3.1+1.1\\2.3+1.2&2.1+1.1\\\end{array}\right]$

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

   $BxB=\left[\begin{array}{ccc}11&4\\8&3\\\end{array}\right]$

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c) (A + B) × (A - B)

   

   · cálculo de A + B

     $A+B=\left[\begin{array}{ccc}2&3\\5&1\\\end{array}\right]+\left[\begin{array}{ccc}3&1\\2&1\\\end{array}\right]$

    $A+B=\left[\begin{array}{ccc}2+3&3+1\\5+2&1+1\\\end{array}\right]$

     $A+B=\left[\begin{array}{ccc}5&4\\7&2\\\end{array}\right]$

   · cálculo de A - B

     $A-B=\left[\begin{array}{ccc}2&3\\5&1\\\end{array}\right]-\left[\begin{array}{ccc}3&1\\2&1\\\end{array}\right]$

     $A-B=\left[\begin{array}{ccc}2-3&3-1\\5-2&1-1\\\end{array}\right]$

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

   · cálculo de (A + B) × (A - B)

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

     $(A+B).(A-B)=\left[\begin{array}{ccc}5.(-1)+4.3&5.2+4.0\\7.(-1)+2.3&7.2+2.0\\\end{array}\right]$

     $(A+B).(A-B)=\left[\begin{array}{ccc}-5+12&10+0\\-7+6&14+0\\\end{array}\right]$

     $(A+B).(A-B)=\left[\begin{array}{ccc}7&10\\-1&14\\\end{array}\right]$