【線形代数】4-3-1 3次行列式の定義｜問題集

\(=\begin{vmatrix}0 & -1 & 0 \\ 5 & 0 & 6 \\ 13 & 4 & 6\end{vmatrix}\)
\(=-\begin{vmatrix}-1 & 0 & 0 \\ 0 & 5 & 6 \\ 13 & 4 & 6\end{vmatrix}\)
\(=-\begin{vmatrix}-1 & 0 & 0 \\ -4 & -8 & 0 \\ 4 & 13 & 6\end{vmatrix}\)
\(=-(-1)\cdot(-8)\cdot6\)
\(=-48\)

\(=-\begin{vmatrix}1 & 0 & 0 \\ 4 & 6 & -11 \\ -2 & 3 & 5\end{vmatrix}\)
\(=1\begin{vmatrix}6 & -11 \\ 3 & 5\end{vmatrix}\)
\(=30-(-33)\)
\(=63\)

\(=\begin{vmatrix}2 & 5 & 0 \\ 3 & 1 & 2 \\ -8 & 1 & 0\end{vmatrix}\)
\(=-\begin{vmatrix}3 & 1 & 2 \\ 2 & 5 & 0 \\ -8 & 1 & 0\end{vmatrix}\)
\(=\begin{vmatrix}2 & 1 & 3 \\ 0 & 5 & 2 \\ 0 & 1 & -8\end{vmatrix}\)
\(=2\begin{vmatrix}5 & 2 \\ 1 & -8\end{vmatrix}\)
\(=2(-40-2)\)
\(=-84\)

\(=\begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ 0 & b^2-ab & c^2-ac\end{vmatrix}\)
\(=\begin{vmatrix}1 & 1 & 1 \\ 0 & b-a & c-a \\ 0 & b(b-a) & c(c-a)\end{vmatrix}\)
\(=1\begin{vmatrix}b-a & c-a \\ b(b-a) & c(c-a)\end{vmatrix}\)
\(=(b-a)(c-a)\begin{vmatrix}1 & 1 \\ b & c\end{vmatrix}\)
\(=(b-a)(c-a)(c-b)\)
\(=(a-b)(b-c)(c-a)\)

\(=\begin{vmatrix}3a+b & a & a \\ 3a+b & a+b & a \\ 3a+b & a & a+b\end{vmatrix}\)
\(=(3a+b)\begin{vmatrix}1 & a & a \\ 1 & a+b & a \\ 1 & a & a+b\end{vmatrix}\)
\(=(3a+b)\begin{vmatrix}1 & a & a \\ 0 & b & 0 \\ 0 & 0 & b\end{vmatrix}\)
\(=(3a+b)\cdot1\cdot b\cdot b\)
\(=b^2(3a+b)\)

\(=a^3+b^3+c^3-acb-bca-cab\)
\(=a^3+b^3+c^3-3abc\)