Calcule a matriz inversa da matriz M= 3 1 3 -2 3 1 3 2 2

Vamos calcular a matriz inversa da matriz M: M = \[ \begin{bmatrix} 3 & 1 & 3 \\ -2 & 3 & 1 \\ 3 & 2 & 2 \\ \end{bmatrix} \] Passo 1: Calcular o determinante de M \[ \det(M) = 3 \times (3 \times 2 - 1 \times 2) - 1 \times (-2 \times 2 - 1 \times 3) + 3 \times (-2 \times 2 - 3 \times 3) \] Calculando: \[ = 3 \times (6 - 2) - 1 \times (-4 - 3) + 3 \times (-4 - 9) \] \[ = 3 \times 4 - 1 \times (-7) + 3 \times (-13) \] \[ = 12 + 7 - 39 = -20 \] Passo 2: Calcular a matriz dos cofatores Calculamos cada cofator \( C_{ij} \): \[ C_{11} = \det \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix} = 3 \times 2 - 1 \times 2 = 6 - 2 = 4 \] \[ C_{12} = -\det \begin{bmatrix} -2 & 1 \\ 3 & 2 \end{bmatrix} = -((-2) \times 2 - 1 \times 3) = -(-4 - 3) = 7 \] \[ C_{13} = \det \begin{bmatrix} -2 & 3 \\ 3 & 2 \end{bmatrix} = (-2) \times 2 - 3 \times 3 = -4 - 9 = -13 \] \[ C_{21} = -\det \begin{bmatrix} 1 & 3 \\ 2 & 2 \end{bmatrix} = - (1 \times 2 - 3 \times 2) = - (2 - 6) = 4 \] \[ C_{22} = \det \begin{bmatrix} 3 & 3 \\ 3 & 2 \end{bmatrix} = 3 \times 2 - 3 \times 3 = 6 - 9 = -3 \] \[ C_{23} = -\det \begin{bmatrix} 3 & 1 \\ 3 & 2 \end{bmatrix} = - (3 \times 2 - 1 \times 3) = - (6 - 3) = -3 \] \[ C_{31} = \det \begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix} = 1 \times 1 - 3 \times 3 = 1 - 9 = -8 \] \[ C_{32} = -\det \begin{bmatrix} 3 & 3 \\ -2 & 1 \end{bmatrix} = - (3 \times 1 - 3 \times (-2)) = - (3 + 6) = -9 \] \[ C_{33} = \det \begin{bmatrix} 3 & 1 \\ -2 & 3 \end{bmatrix} = 3 \times 3 - 1 \times (-2) = 9 + 2 = 11 \] Matriz dos cofatores: \[ \begin{bmatrix} 4 & 7 & -13 \\ 4 & -3 & -3 \\ -8 & -9 & 11 \\ \end{bmatrix} \] Passo 3: Calcular a matriz adjunta (transposta da matriz dos cofatores) \[ \text{adj}(M) = \begin{bmatrix} 4 & 4 & -8 \\ 7 & -3 & -9 \\ -13 & -3 & 11 \\ \end{bmatrix} \] Passo 4: Calcular a inversa \[ M^{-1} = \frac{1}{\det(M)} \times \text{adj}(M) = -\frac{1}{20} \times \begin{bmatrix} 4 & 4 & -8 \\ 7 & -3 & -9 \\ -13 & -3 & 11 \\ \end{bmatrix} \] Ou seja, \[ M^{-1} = \begin{bmatrix} -\frac{4}{20} & -\frac{4}{20} & \frac{8}{20} \\ -\frac{7}{20} & \frac{3}{20} & \frac{9}{20} \\ \frac{13}{20} & \frac{3}{20} & -\frac{11}{20} \\ \end{bmatrix} = \begin{bmatrix} -\frac{1}{5} & -\frac{1}{5} & \frac{2}{5} \\ -\frac{7}{20} & \frac{3}{20} & \frac{9}{20} \\ \frac{13}{20} & \frac{3}{20} & -\frac{11}{20} \\ \end{bmatrix} \] Resposta final: \[ M^{-1} = \begin{bmatrix} -0,2 & -0,2 & 0,4 \\ -0,35 & 0,15 & 0,45 \\ 0,65 & 0,15 & -0,55 \\ \end{bmatrix} \]