Calcule a inversa das matrizes usando operações elementares.
\(\left[\begin{array}{ccc}2 & 2 & -4 \\\\ 2 & 2 & 1 \\\\ 2 & -8 & 1\end{array}\right])

Passo 1

O enunciado pede para calcular a inversa da matriz
A = \begin{pmatrix} 2 & 2 & -4\\\\ 2 & 2 & 1\\\\ 2 & -8 & 1 \end{pmatrix}
usando operações elementares (Gauss–Jordan). Vamos montar a matriz aumentada [A\mid I] e transformá-la até obter [I\mid A^{-1}].

Passo 2

Montamos a matriz aumentada e eliminamos no segundo e terceiro lugares da primeira coluna:
\left[\begin{array}{ccc|ccc} 2 & 2 & -4 & 1 & 0 & 0\\\\ 2 & 2 & 1 & 0 & 1 & 0\\\\ 2 & -8& 1 & 0 & 0 & 1 \end{array}\right] \quad R_2\leftarrow R_2 - R_1,\; R_3\leftarrow R_3 - R_1 \;\longrightarrow\; \left[\begin{array}{ccc|ccc} 2 & 2 & -4 & 1 & 0 & 0\\\\ 0 & 0 & 5 & -1 & 1 & 0\\\\ 0 & -10 & 5 & -1 & 0 & 1 \end{array}\right]

Passo 3

Trocamos R_2 e R_3 para obter pivô não nulo em coluna 2, e depois escalonamos cada linha para pivô 1:
\begin{aligned} &R_2\leftrightarrow R_3:\\\ &\quad\left[\begin{array}{ccc|ccc} 2 & 2 & -4 & 1 & 0 & 0\\\\ 0 & -10 & 5 & -1 & 0 & 1\\\\ 0 & 0 & 5 & -1 & 1 & 0 \end{array}\right]\\\ &R_1\leftarrow \tfrac12R_1,\; R_2\leftarrow -\tfrac1{10}R_2,\; R_3\leftarrow \tfrac15R_3: \; \left[\begin{array}{ccc|ccc} 1 & 1 & -2 & \tfrac12 & 0 & 0\\\\ 0 & 1 & -\tfrac12 & \tfrac1{10}& 0 & -\tfrac1{10}\\\\ 0 & 0 & 1 & -\tfrac15 & \tfrac15 & 0 \end{array}\right] \end{aligned}

Passo 4

Eliminamos os não-zeros acima dos pivôs em colunas 2 e 3:
\begin{aligned} &R_1\leftarrow R_1 - R_2:\; \left[\begin{array}{ccc|ccc} 1 & 0 & -\tfrac32 & \tfrac25 & 0 & \tfrac1{10}\\\\ 0 & 1 & -\tfrac12 & \tfrac1{10} & 0 & -\tfrac1{10}\\\\ 0 & 0 & 1 & -\tfrac15 & \tfrac15 & 0 \end{array}\right],\\\ &R_1\leftarrow R_1 + \tfrac32\,R_3,\; R_2\leftarrow R_2 + \tfrac12\,R_3: \quad \left[\begin{array}{ccc|ccc} 1 & 0 & 0 & \tfrac1{10} & \tfrac3{10} & \tfrac1{10}\\\\ 0 & 1 & 0 & 0 & \tfrac1{10} & -\tfrac1{10}\\\\ 0 & 0 & 1 & -\tfrac15 & \tfrac15 & 0 \end{array}\right]. \end{aligned}

Resposta

A inversa de A é
A^{-1} = \begin{pmatrix} \tfrac{1}{10} & \tfrac{3}{10} & \tfrac{1}{10}\\[6pt] 0 & \tfrac{1}{10} & -\tfrac{1}{10}\\[6pt] -\tfrac{1}{5} & \tfrac{1}{5} & 0 \end{pmatrix}.]

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