Category:Expansion Theorem for Determinants

This category contains examples of use of Expansion Theorem for Determinants.

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $D = \map \det {\mathbf A}$ be the determinant of $\mathbf A$:

$\ds \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_\lambda \map \sgn \lambda a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$

where:

the summation $\ds \sum_\lambda$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$

$\map \sgn \lambda$ is the sign of the permutation $\lambda$.

Let $a_{p q}$ be an element of $\mathbf A$.

Let $A_{p q}$ be the cofactor of $a_{p q}$ in $D$.

Then:

$(1): \quad \ds \forall r \in \closedint 1 n: D = \sum_{k \mathop = 1}^n a_{r k} A_{r k}$

$(2): \quad \ds \forall c \in \closedint 1 n: D = \sum_{k \mathop = 1}^n a_{k c} A_{k c}$

Thus the value of a determinant can be found either by:

multiplying all the elements in a row by their cofactors and adding up the products

or:

multiplying all the elements in a column by their cofactors and adding up the products.

The identity:

$\ds D = \sum_{k \mathop = 1}^n a_{r k} A_{r k}$

is known as the expansion of $D$ in terms of row $r$, while:

$\ds D = \sum_{k \mathop = 1}^n a_{k c} A_{k c}$

is known as the expansion of $D$ in terms of column $c$.