Questão 4.

Dadas as matrizes A e B abaixo, determine os produtos A B e B A :
A=\left[\begin{array}{ll} 2 & 1 \\\\ 1 & 0 \\\\ 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{lll} 1 & 0 & 1 \\\\ 0 & 1 & 1 \end{array}\right] .

Passo 1

O enunciado nos dá
A\in\mathbb{R}^{3\times2},\quad B\in\mathbb{R}^{2\times3}
e pede os produtos AB e BA. Para isso vamos usar a definição de multiplicação de matrizes:
(AB)_{ij}=\sum_{k=1}^{2}A_{ik}\,B_{kj}, \qquad (BA)_{ij}=\sum_{k=1}^{3}B_{ik}\,A_{kj}.

Passo 2

Calcular AB\in\mathbb{R}^{3\times3}. Usando (AB)_{ij}=\sum_{k=1}^{2}A_{ik}B_{kj}:
AB=\begin{pmatrix} 2\cdot1+1\cdot0 & 2\cdot0+1\cdot1 & 2\cdot1+1\cdot1\\\\ 1\cdot1+0\cdot0 & 1\cdot0+0\cdot1 & 1\cdot1+0\cdot1\\\\ 0\cdot1+1\cdot0 & 0\cdot0+1\cdot1 & 0\cdot1+1\cdot1 \end{pmatrix} =\begin{pmatrix} 2 & 1 & 3\\\\ 1 & 0 & 1\\\\ 0 & 1 & 1 \end{pmatrix}.

Passo 3

Calcular BA\in\mathbb{R}^{2\times2}. Usando (BA)_{ij}=\sum_{k=1}^{3}B_{ik}A_{kj}:
BA=\begin{pmatrix} 1\cdot2+0\cdot1+1\cdot0 & 1\cdot1+0\cdot0+1\cdot1\\\\ 0\cdot2+1\cdot1+1\cdot0 & 0\cdot1+1\cdot0+1\cdot1 \end{pmatrix} =\begin{pmatrix} 2 & 2\\\\ 1 & 1 \end{pmatrix}.

Resposta

AB=\begin{pmatrix} 2 & 1 & 3\\\\ 1 & 0 & 1\\\\ 0 & 1 & 1 \end{pmatrix}, \qquad BA=\begin{pmatrix} 2 & 2\\\\ 1 & 1 \end{pmatrix}.

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