Category:Laplace's Expansion Theorem

This category contains pages concerning Laplace's Expansion Theorem:

Let $D$ be the determinant of order $n$.

Let $r_1, r_2, \ldots, r_k$ be integers such that:

$1 \le k < n$

$1 \le r_1 < r_2 < \cdots < r_k \le n$

Let $\map D {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$ be an order-$k$ minor of $D$.

Let $\map {\tilde D} {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$ be the cofactor of $\map D {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$.

Then:

$\ds D = \sum_{1 \mathop \le u_1 \mathop < \cdots \mathop < u_k \mathop \le n} \map D {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k} \, \map {\tilde D} {r_1, r_2, \ldots, r_k \mid u_1, u_2, \ldots, u_k}$

A similar result applies for columns.

Source of Name

This entry was named for Pierre-Simon de Laplace.