Question

São dadas as matrizes A= [2 1][3 1] e B [3 4][1 0]
a) calcule AxB
b) calcule BxA
c) calcule A²

Answer 1

Olá

A multiplicação de matrizes é feito através de linhas por colunas


A)

$A \left[\begin{array}{ccc}2&1\\3&1\\\end{array}\right]\cdot B \left[\begin{array}{ccc}3&4\\1&0\\\end{array}\right] \\ \\ \\ A\cdot B= \left[\begin{array}{ccc}((2\cdot3)+(1\cdot1))~~~~&((2\cdot4)+(1\cdot0))\\((3\cdot3)+(1\cdot1))~~~~&((3\cdot4)+(1\cdot0))\\\end{array}\right] \\ \\ \\ A\cdot B= \left[\begin{array}{ccc}(6+1)~~~~&(8+0)\\(9+1)~~~~&(12+0)\\\end{array}\right] \\ \\ \\ \boxed{\boxed{A\cdot B= \left[\begin{array}{ccc}7&8\\10&12\\\end{array}\right]}}$




B)

$B \left[\begin{array}{ccc}3&4\\1&0\\\end{array}\right] \cdot A \left[\begin{array}{ccc}2&1\\3&1\\\end{array}\right] \\ \\ \\ B\cdot A= \left[\begin{array}{ccc}((3\cdot2)+(4\cdot3))~~~~&((3\cdot1)+(4\cdot1))\\((1\cdot2)+(0\cdot3))~~~~&((1\cdot1)+(0\cdot1))\\\end{array}\right] \\ \\ \\ B\cdot A= \left[\begin{array}{ccc}(6+12)~~~~&(3+4)\\(2+0)~~~~&(1+0)\\\end{array}\right] \\ \\ \\ \boxed{\boxed{B\cdot A= \left[\begin{array}{ccc}18&7\\2&1\\\end{array}\right]}}$



C)

Podemos obter a matriz A², multiplicando A por A. então  A²=A*A;

$A^2=A \left[\begin{array}{ccc}2&1\\3&1\\\end{array}\right]\cdot A\left[\begin{array}{ccc}2&1\\3&1\\\end{array}\right] \\ \\ \\ A^2= \left[\begin{array}{ccc}((2\cdot2)+(1\cdot3))~~~~&((2\cdot1)+(1\cdot1))\\((3\cdot2)+(1\cdot3))~~~~&((3\cdot1)+(1\cdot1))\\\end{array}\right] \\ \\ \\ A^2= \left[\begin{array}{ccc}(4+3)~~~~&(2+1)\\(6+3)~~~~&(3+1)\\\end{array}\right] \\ \\ \\ \boxed{\boxed{A^2= \left[\begin{array}{ccc}7&3\\9&4\\\end{array}\right]}}$





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