Question

Determinar a inversa( Mostrar processos feitos por favor

Answer 1

Olá

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

1º vamos ver se realmente a matriz possui inversa.
Para isso, basta calcular o determinante, se der diferente de zero, então a matriz possui inversa

Por sarrus

$\displaystyle A=\left|\begin{array}{ccc}-1&-1&2\\2&1&-2\\1&1&-1\end{array}\right|\\\\\\\mathsf{det~A=\underbrace{(\mathsf{1+2+4})}_{diag.~principal}-\underbrace{(\mathsf{2+2+2})}_{diag.~secund\'aria}}\\\\\\\mathsf{det ~A=7-6}\\\\\mathsf{det~A=1}$

A matriz possui inversa.

Podemos calcular a inversa de uma matriz com a seguinte fórmula

$\displaystyle \mathsf{A^{-1}= \frac{1}{det~A}~\cdot~(cof(A) )^T}$


Já temos o determinante, então vamos encontrar a matriz dos cofatores de A.

$\displaystyle \mathsf{C_{11}=(-1)^{1+1}}\cdot \left|\begin{array}{ccc}1&-2\\1&-1\\\end{array}\right| ~=~(-1)^2\cdot (-1+2)~=~\boxed{1}\\\\\\ \mathsf{C_{12}=(-1)^{1+2}}\cdot \left|\begin{array}{ccc}2&-2\\1&-1\\\end{array}\right| ~=~(-1)^3\cdot (-2+2)~=~\boxed{0}\\\\\\ \mathsf{C_{13}=(-1)^{1+3}}\cdot \left|\begin{array}{ccc}2&1\\1&1\\\end{array}\right| ~=~(-1)^4\cdot (2-1)~=~\boxed{1}$

$\displaystyle \mathsf{C_{21}=(-1)^{2+1}}\cdot \left|\begin{array}{ccc}-1&2\\1&-1\\\end{array}\right| ~=~(-1)^3\cdot (1-2)~=~\boxed{1}\\\\\\ \mathsf{C_{22}=(-1)^{2+2}}\cdot \left|\begin{array}{ccc}-1&2\\1&-1\\\end{array}\right| ~=~(-1)^4\cdot (1-2)~=~\boxed{-1}\\\\\\ \mathsf{C_{23}=(-1)^{2+3}}\cdot \left|\begin{array}{ccc}-1&-1\\1&1\\\end{array}\right| ~=~(-1)^5\cdot (-1+1)~=~\boxed{0}$

$\displaystyle \mathsf{C_{31}=(-1)^{3+1}}\cdot \left|\begin{array}{ccc}-1&2\\1&-2\\\end{array}\right| ~=~(-1)^4\cdot (2-2)~=~\boxed{0}\\\\\\ \mathsf{C_{32}=(-1)^{3+2}}\cdot \left|\begin{array}{ccc}-1&2\\2&-2\\\end{array}\right| ~=~(-1)^5\cdot (2-4)~=~\boxed{2}\\\\\\ \mathsf{C_{33}=(-1)^{3+3}}\cdot \left|\begin{array}{ccc}-1&-1\\2&1\\\end{array}\right| ~=~(-1)^6\cdot (-1+2)~=~\boxed{1}$

A matriz dos cofatores ficou dessa forma

$\mathsf{cof(A)= \left[\begin{array}{ccc}1&0&1\\1&-1&0\\0&2&1\end{array}\right] }$


Agora, para encontrar a matriz transposta, basta colocar tudo que está em linha, em colunas, e o que estiver em colunas, colocá-las em linhas.

$\displaystyle \mathsf{(cof(A))^T= \left[\begin{array}{ccc}1&1&0\\0&-1&2\\1&0&1\end{array}\right] }$


A nossa fórmula para calcular a inversa

$\displaystyle \mathsf{A^{-1}= \frac{1}{det~A}~\cdot~(cof(A) )^T}$

Substituindo

$\displaystyle \mathsf{A^{-1}= \frac{1}{1}~\cdot~\left[\begin{array}{ccc}1&1&0\\0&-1&2\\1&0&1\end{array}\right]}\\\\\\\boxed{\mathsf{A^{-1}=\left[\begin{array}{ccc}1&1&0\\0&-1&2\\1&0&1\end{array}\right]}}\qquad \qquad\Longleftarrow \text{Resposta}$


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