Matemática, perguntado por fmklingelfus, 8 meses atrás

Encontre a inversa de uma matriz 2x2

Encontre a matriz inversa

B= [ 3 6 ]
[ -10 -10]

Soluções para a tarefa

Respondido por Nasgovaskov

Nesta questão, se pede para encontrar a inversa da matriz B (2x2)

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$\begin{array}{l}\sf B=\begin{bmatrix}~~\sf3&~~\sf6\\\sf-10&~~\sf-10~~\end{bmatrix}\end{array}$

Primeiro calcule o determinante, para isso multiplique uma diagonal (principal) e subtraia da multiplicação de outra diagonal (secundária):

$\begin{array}{l}\sf Det(B)=\begin{vmatrix}~~\sf3&~~\sf6\\\sf-10&~~\sf-10~~\end{vmatrix}\\\\\sf Det(B)=3\cdot(-10)-[6\cdot(-10)]\\\\\sf Det(B)=-30-[-60]\\\\\sf Det(B)=-30+60\\\\\!\boxed{\sf Det(B)=30}\end{array}$

Agora divida cada elemento da matriz B pelo determinante:

$\begin{array}{l}\sf B=\begin{bmatrix}~~\sf\dfrac{3}{30}&~~\sf\dfrac{6}{30}\\\\\sf-\dfrac{10}{30}&~~\sf-\dfrac{10}{30}~~\end{bmatrix}\\\\\sf \Rightarrow~~Simplifique:\\\\\sf B=\begin{bmatrix}~~\sf\dfrac{3^{\,:3}}{30^{\,:3}}&~~\sf\dfrac{6^{\,:6}}{30^{\,:6}}\\\\\sf-\dfrac{10^{\,:10}}{30^{\,:10}}&~~\sf-\dfrac{10^{\,:10}}{30^{\,:10}}~~\end{bmatrix}\\\\ \sf B=\begin{bmatrix}~~\sf\dfrac{1}{10}&~~\sf\dfrac{1}{5}\\\\\sf-\dfrac{1}{3}&~~\sf-\dfrac{1}{3}~~\end{bmatrix}\end{array}$

Troque os elementos da diagonal principal:

$\begin{array}{l}\sf B=\begin{bmatrix}\sf-\dfrac{1}{3}~~&\sf\dfrac{1}{5}~\\\\\sf-\dfrac{1}{3}~&\sf\dfrac{1}{10}~~\end{bmatrix}\end{array}$

Por fim inverta os sinais da diagonal secundária:

$\begin{array}{l}\sf B=\begin{bmatrix}\sf-\dfrac{1}{3}~~&\sf - \dfrac{1}{5}~~\\\\\sf\dfrac{1}{3}&\sf\dfrac{1}{10}\end{bmatrix}\end{array}$

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Resposta final:

↓Assim, esta é a matriz inversa de B↓

$\boxed{\boxed{\begin{array}{l} \\ \: \: \: \boldsymbol{\sf B^{-1}=}\begin{bmatrix}\boldsymbol{\sf-\dfrac{1}{3}}~~&\boldsymbol{\sf - \dfrac{1}{5}}~~\\\\\boldsymbol{\sf\dfrac{1}{3}}&\boldsymbol{\sf\dfrac{1}{10}}\end{bmatrix}\quad \\ \\ \end{array}}}$