Macierze, Rozwiniecie LaPlace'a

[domino21]

c.

\(\begin{vmatrix} (1+x) \ 1 \ 1 \ 1 \\ 1 \ (1-x) \ 1 \ 1\\ 1 \ 1 \ (1+x) \ 1 \\ 1 \ 1 \ 1 \ (1-x)\end{vmatrix} =0
\begin{vmatrix} (1+x) \ x \ x \ 0 \\ 1 \ x \ 0 \ 0 \\ 1 \ 0 \ -x \ 0 \\ 1 \ 0 \ 0 \ x \end{vmatrix}=0
x \begin{vmatrix} (1+x) \ x \ x \\ 1 \ x \ 0 \\ 1 \ 0 \ -x \end{vmatrix}=0
x \begin{vmatrix} (1+x) \ x \ x \\ 1 \ x \ 0 \\ 2+x \ x \ 0 \end{vmatrix}=0
x^2 \begin{vmatrix} 1 \ \ \ \ \ x \\ (2+x) \ x\end{vmatrix}=0
x^3 \begin{vmatrix} 1 \ \ \ \ \ \ \ 1 \\ (2+x) \ 1\end{vmatrix}=0
x^3(1-2-x)=0
x^3(x+1)=0
x=-1 \ \vee \ x=0\)