[tex]A(a)=\begin{pmatrix}
a& 1 &1 \\
1& a & 1\\
1& 1 & a
\end{pmatrix} [/tex]
[tex]A( - 1)=\begin{pmatrix}
- 1& 1 &1 \\
1& - 1 & 1\\
1& 1 & - 1
\end{pmatrix}[/tex]
[tex]detA(-1)=\begin{vmatrix}
-1& 1 & 1\\
1& -1 & 1\\
1& 1& -1
\end{vmatrix} [/tex]
[tex]detA(-1)=-1 \times (-1) \times (-1)+1 \times 1 \times 1+1 \times 1 \times 1-1 \times (-1) \times 1-1 \times 1 \times (-1)-(-1) \times 1 \times 1
[/tex]
[tex]detA( - 1)=-1+1+1+1+1+1
[/tex]
[tex]detA( - 1) = 4 \: \neq \: 0 = > A( - 1) \:este \: inversabila=>\exists\:{A(-1)}^{-1}[/tex]
[tex]{A(-1)}^{t}=\begin{pmatrix}
- 1& 1& 1\\
1 & - 1& 1\\
1& 1 & - 1
\end{pmatrix} = {A(-1)}^{t}=A( - 1)[/tex]
[tex] A(-1)_{11}={(-1)}^{1+1}\times\begin{vmatrix}
- 1& 1\\
1 & - 1
\end{vmatrix} = 1 \times (1 - 1) = 1 \times 0 = 0[/tex]
[tex]A(-1)_{12}={(-1)}^{1+2}\times\begin{vmatrix}
1&1 \\
1& - 1
\end{vmatrix} = - 1 \times ( - 2) = 2[/tex]
[tex]A(-1)_{13}={(-1)}^{1+3}\times\begin{vmatrix}
1& - 1\\
1 & 1
\end{vmatrix} = 1 \times( 1 + 1) = 1 \times 2 = 2[/tex]
[tex]A(-1)_{21}={(-1)}^{2+1}\times\begin{vmatrix}
1& 1\\
1& - 1
\end{vmatrix} = - 1 \times ( - 1 - 1) = - 1 \times ( - 2) = 2[/tex]
[tex]A(-1)_{22}={(-1)}^{2+2}\times\begin{vmatrix}
- 1&1 \\
1& - 1
\end{vmatrix} = 1 \times (1 - 1) = 1 \times 0 = 0[/tex]
[tex]A(-1)_{23}={(-1)}^{2+3}\times\begin{vmatrix}
- 1& 1\\
1 & 1
\end{vmatrix} = - 1 \times ( - 1 - 1) = - 1 \times ( - 2) = 2[/tex]
[tex]A(-1)_{31}={(-1)}^{3+1}\times\begin{vmatrix}
1& 1\\
- 1& 1
\end{vmatrix} = 1 \times (1 + 1) = 1 \times 2 = 2[/tex]
[tex]A(-1)_{32}={(-1)}^{3+2}\times\begin{vmatrix}
- 1&1 \\
1& 1
\end{vmatrix} = - 1 \times ( - 1 - 1) = - 1 \times ( - 2) = 2[/tex]
[tex]A(-1)_{33}={(-1)}^{3+3}\times\begin{vmatrix}
- 1& 1\\
1& - 1
\end{vmatrix} = 1 \times (1 - 1) = 1 \times 0 = 0[/tex]
[tex]{A(-1)}^{*}=\begin{pmatrix}
A(-1)_{11}& A(-1)_{12} &A(-1)_{13} \\
A(-1)_{21} & A(-1)_{22} &A(-1)_{23} \\
A(-1)_{31} &A(-1)_{32} & A(-1)_{33}
\end{pmatrix} [/tex]
[tex]{A(-1)}^{*} = \begin{pmatrix}
0& 2 & 2\\
2& 0 &2 \\
2& 2 & 0
\end{pmatrix} [/tex]
[tex]{A(-1)}^{-1}=\frac{1}{detA(-1)}\times{A(-1)}^{*} [/tex]
[tex]{A(-1)}^{-1} = \frac{1}{4} \begin{pmatrix}
0& 2 & 2\\
2& 0 &2 \\
2& 2 & 0
\end{pmatrix} [/tex]
[tex]{A(-1)}^{-1} = \begin{pmatrix}
0& \frac{2}{4} & \frac{2}{4} \\
\frac{2}{4} & 0 & \frac{2}{4} \\
\frac{2}{4} & \frac{2}{4} & 0
\end{pmatrix} [/tex]
[tex]{A(-1)}^{-1} = \begin{pmatrix}
0& \frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & 0 & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2} & 0
\end{pmatrix} [/tex]
