Laplace Expansion
# Laplace Expansion
$$ A = a_{ij} $$
$$ 1 \le i_1 \le .. \le i_r \le n $$
$$ 1 \le j_1 \le .. \le j_r \le n $$
definition : r-dim. small matrix of A
$$ A\binom{i_1 .. i_r}{j_1 .. j_r} \equiv
\begin{pmatrix}
a_{i_1,j_1}&\cdots&a_{i_1,j_r}\\
\vdots&\ddots&\vdots\\
a_{i_r,j_1}&\cdots&a_{i_r,j_r}\\
\end{pmatrix}
$$
we define the rest elements $ i^\prime, j^\prime $
$$ \{ i_1^{\prime}, .. , i^\prime_{n-r} \} \equiv \{ 1 .. n \} \setminus \{ i_1 , .. , i_r \} $$
$$ \{ j_1^{\prime}, .. , j^\prime_{n-r} \} \equiv \{ 1 .. n \} \setminus \{ j_1 , .. , j_r \} $$
where,
$$ 1 \le i^\prime_1 \le .. \le i^\prime_{n-r} \le n $$
$$ 1 \le j^\prime_1 \le .. \le j^\prime_{n-r} \le n $$
Here, we can express Determinant A :
$$ detA = \sum_{j_1 \le .. \le jr} (-1)^{i_1 + .. + i_r + j_1 + .. + j_r}
\begin{vmatrix}
a_{i_1,j_1}&\cdots&a_{i_1,j_r}\\
\vdots&\ddots&\vdots\\
a_{i_r,j_1}&\cdots&a_{i_r,j_r}\\
\end{vmatrix}
\begin{vmatrix}
a_{i^\prime_1,j^\prime_1}&\cdots&a_{i^\prime_1,j^\prime_{n-r}}\\
\vdots&\ddots&\vdots\\
a_{i^\prime_{n-r},j^\prime_1}&\cdots&a_{i^\prime_{n-r},j^\prime_{n-r}}\\
\end{vmatrix}
$$
# NOTE : implementation
if we define determinant in program ,
we should avoid this way .
Because even determinant of 10x10 matrix costs much time .
we should use determinant of "Triangled Matrix" .