Concursos Públicos ⇒ Matriz Tópico resolvido

[japaesah]

A matriz inversa da matriz A é
[tex3]A^{-1}=[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]A^{-1}=[/tex3]

[tex3]\begin{pmatrix}
16 & -1 & -10 \\
13 & -1 & -8 \\
11 & -1 & -7 \\
\end{pmatrix}[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\begin{pmatrix}
16 & -1 & -10 \\
13 & -1 & -8 \\
11 & -1 & -7 \\
\end{pmatrix}[/tex3]

Lembrando que A .[tex3]A^{-1} = I_{3}[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]A^{-1} = I_{3}[/tex3]

denota a matriz identidade de ordem 3 , a segunda linha de A é

A) ( 1 1 1 )
B) ( 2 1 −3 )
C) ( 3 −2 −2 )
D) ( 0 0 −1 )
E) ( 2 −2 3 )


Alguma Vídeo-aula com exemplos para indicar? Obg

Resposta

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ALTERNATIVA C

[petras]

[tex3]\begin{pmatrix}
16 & -1 & -10 ~|1&0 &0\\
13 & -1 & -8 ~~~|0&1&0\\
11 & -1 & -7 ~ ~~|0&0&1\\
\end{pmatrix}L1.\frac{1}{16}\rightarrow \begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} ~~~~|\frac{1}{16}&0 &0\\
13 & -1 & -8 ~~~|0&1&0\\
11 & -1 & -7 ~ ~~|0&0&1\\
\end{pmatrix}-13L1+L2; -11L1 + L3\rightarrow \begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} |~~\frac{1}{16}&0 &0\\
0 & -\frac{3}{16}& \frac{1}{8} |-\frac{13}{16}&1&0\\
0 & -\frac{5}{16} & -\frac{1}{8} |-\frac{11}{16}&0&1\\
\end{pmatrix}L2.-\frac{16}{3}\rightarrow \\ \begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} |~~\frac{1}{16}&0 &0\\
0 & 1& -\frac{2}{3} |\frac{13}{3}&-\frac{16}{3}&0\\
0 & -\frac{5}{16} & -\frac{1}{8} |-\frac{11}{16}&0&1\\
\end{pmatrix}L2.\frac{5}{16}+L3\rightarrow
\begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} |~~\frac{1}{16}&0 &0\\
0 & 1& -\frac{2}{3} |\frac{13}{3}&1&0\\
0 & 0 & -\frac{1}{3} |\frac{2}{3}&-\frac{5}{3}&1\\
\end{pmatrix}-3L3\rightarrow \\
\begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} |~~\frac{1}{16}&0 &0\\
0 & 1& -\frac{2}{3} |\frac{13}{3}&1&0\\
0 & 0 & 1 |-2&5&-3\\
\end{pmatrix}\frac{2}{3}.L3+L2;\frac{5}{8}.L3+L1\rightarrow \begin{pmatrix}
1 & -\frac{1}{16} & 0 |~~-\frac{19}{16}&\frac{25}{8} &-\frac{15}{8}\\
0 & 1& 0 |3&-2&-2\\
0 & 0 & 1 |-2&5&-3\\
\end{pmatrix}\frac{1}{16}.L2+L1\rightarrow\begin{pmatrix}
1 & 0 & 0 |~~-1&3 &-2\\
0 & 1& 0 |3&-2&-2\\
0 & 0 & 1 |-2&5&-3\\
\end{pmatrix}[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\begin{pmatrix}
16 & -1 & -10 ~|1&0 &0\\
13 & -1 & -8 ~~~|0&1&0\\
11 & -1 & -7 ~ ~~|0&0&1\\
\end{pmatrix}L1.\frac{1}{16}\rightarrow \begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} ~~~~|\frac{1}{16}&0 &0\\
13 & -1 & -8 ~~~|0&1&0\\
11 & -1 & -7 ~ ~~|0&0&1\\
\end{pmatrix}-13L1+L2; -11L1 + L3\rightarrow \begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} |~~\frac{1}{16}&0 &0\\
0 & -\frac{3}{16}& \frac{1}{8} |-\frac{13}{16}&1&0\\
0 & -\frac{5}{16} & -\frac{1}{8} |-\frac{11}{16}&0&1\\
\end{pmatrix}L2.-\frac{16}{3}\rightarrow \\ \begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} |~~\frac{1}{16}&0 &0\\
0 & 1& -\frac{2}{3} |\frac{13}{3}&-\frac{16}{3}&0\\
0 & -\frac{5}{16} & -\frac{1}{8} |-\frac{11}{16}&0&1\\
\end{pmatrix}L2.\frac{5}{16}+L3\rightarrow
\begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} |~~\frac{1}{16}&0 &0\\
0 & 1& -\frac{2}{3} |\frac{13}{3}&1&0\\
0 & 0 & -\frac{1}{3} |\frac{2}{3}&-\frac{5}{3}&1\\
\end{pmatrix}-3L3\rightarrow \\
\begin{pmatrix}
1 & -\frac{1}{16} & -\frac{5}{8} |~~\frac{1}{16}&0 &0\\
0 & 1& -\frac{2}{3} |\frac{13}{3}&1&0\\
0 & 0 & 1 |-2&5&-3\\
\end{pmatrix}\frac{2}{3}.L3+L2;\frac{5}{8}.L3+L1\rightarrow \begin{pmatrix}
1 & -\frac{1}{16} & 0 |~~-\frac{19}{16}&\frac{25}{8} &-\frac{15}{8}\\
0 & 1& 0 |3&-2&-2\\
0 & 0 & 1 |-2&5&-3\\
\end{pmatrix}\frac{1}{16}.L2+L1\rightarrow\begin{pmatrix}
1 & 0 & 0 |~~-1&3 &-2\\
0 & 1& 0 |3&-2&-2\\
0 & 0 & 1 |-2&5&-3\\
\end{pmatrix}[/tex3]