Dadas as matrizes A=\left[\begin{array}{cccc}1 & -5 & -1 & 2 \\\\ 0 & 2 & -3 & 4 \\\\ 0 & 0 & 4 & -2 \\\\ 0 & 0 & 0 & 3\end{array}\right] e B=\left[\begin{array}{cccc}-3 & 0 & 0 & 0 \\\\ 3 & -4 & 0 & 0 \\\\ 2 & 2 & -1 & 0 \\\\ 2 & 1 & 1 & -2\end{array}\right], determine:
A^{-1};

Passo 1

O enunciado pede A^{-1}\) para \[A=\begin{pmatrix} 1 & -5 & -1 & 2 \\\\ 0 & 2 & -3 & 4 \\\\ 0 & 0 & 4 & -2 \\\\ 0 & 0 & 0 & 3 \end{pmatrix}
Sabemos que A é triangular superior com diagonais não nulas, logo A^{-1} também é triangular superior. Vamos usar a fórmula
n_{ij} = \begin{cases} 1/a_{ii},&i=j,\\\\ -\frac{1}{a_{ii}}\sum_{k=i+1}^j a_{ik}\,n_{kj},&i<j,\\\\ 0,&i>j, \end{cases}
onde n_{ij}\] são os elementos de \(A^{-1}.

Passo 2

Calculamos primeiro as entradas diagonais de A^{-1}:
n_{11}=\frac1{1}=1,\quad n_{22}=\frac1{2}=\tfrac12,\quad n_{33}=\frac1{4}=\tfrac14,\quad n_{44}=\frac1{3}=\tfrac13.

Passo 3

Calculamos a única entrada não nula na linha 3 acima da diagonal:
n_{34}=-\frac{1}{a_{33}}\bigl(a_{34}\,n_{44}\bigr) =-\frac{1}{4}\bigl((-2)\cdot\tfrac13\bigr) =\frac{1}{6}.

Passo 4

Agora as entradas da linha 2 acima da diagonal:
n_{23}=-\frac{1}{a_{22}}\bigl(a_{23}\,n_{33}\bigr) =-\frac{1}{2}\bigl((-3)\cdot\tfrac14\bigr) =\frac{3}{8},
n_{24}=-\frac{1}{a_{22}}\bigl(a_{23}\,n_{34}+a_{24}\,n_{44}\bigr) =-\frac{1}{2}\Bigl((-3)\cdot\tfrac16+4\cdot\tfrac13\Bigr) =-\frac{5}{12}.

Passo 5

Finalmente as entradas da linha 1 acima da diagonal:
n_{12}=-\frac{1}{a_{11}}\bigl(a_{12}\,n_{22}\bigr) =-1\cdot\bigl((-5)\cdot\tfrac12\bigr) =\frac{5}{2},
n_{13}=-\frac{1}{a_{11}}\bigl(a_{12}\,n_{23}+a_{13}\,n_{33}\bigr) =-\bigl((-5)\cdot\tfrac{3}{8}+(-1)\cdot\tfrac14\bigr) =\frac{17}{8},
n_{14}=-\frac{1}{a_{11}}\bigl(a_{12}\,n_{24}+a_{13}\,n_{34}+a_{14}\,n_{44}\bigr) =-\Bigl((-5)\cdot\bigl(-\tfrac{5}{12}\bigr)+(-1)\cdot\tfrac16+2\cdot\tfrac13\Bigr) =-\frac{31}{12}.

Resposta

A^{-1} = \begin{pmatrix} 1 & \tfrac{5}{2} & \tfrac{17}{8} & -\tfrac{31}{12} \\[6pt] 0 & \tfrac12 & \tfrac{3}{8} & -\tfrac{5}{12} \\[6pt] 0 & 0 & \tfrac14 & \tfrac16 \\[6pt] 0 & 0 & 0 & \tfrac13 \end{pmatrix}.]

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