Suppose
\(
A=
\begin{bmatrix}
a_{1} & a_{2} & a_{3}
\end{bmatrix}
, a_i\in V, i=\{1,2,3\}
\)
Key idea
\(
\begin{bmatrix}
a_{1} & a_{2} & a_{3}
\end{bmatrix}
\begin{bmatrix}
1 & x_1 & 0 \\
0 & x_2 & 0 \\
0 & x_3 & 1
\end{bmatrix}
=
\begin{bmatrix}
a_{1} & b & a_3
\end{bmatrix}
=
B_2
\)
\(
\Rightarrow
\det(
\begin{bmatrix}
a_{1} & a_{2} & a_{3}
\end{bmatrix}
\begin{bmatrix}
1 & x_1 & 0 \\
0 & x_2 & 0 \\
0 & x_3 & 1
\end{bmatrix}
)
=
\det(B_2)
\)
\(
\Rightarrow
\det(A)
\det(
\begin{bmatrix}
1 & x_1 & 0 \\
0 & x_2 & 0 \\
0 & x_3 & 1
\end{bmatrix}
)
=
\det(B_2)
\)
\(
\Rightarrow
\det(
\begin{bmatrix}
1 & x_1 & 0 \\
0 & x_2 & 0 \\
0 & x_3 & 1
\end{bmatrix}
)
=x_2
=
\frac{\det(B_2)}{\det(A)}
\)
以此類推。
\(
x_1=\frac{\det B_1}{\det A}
\)
\(
x_3=\frac{\det B_3}{\det A}
\)