Category:Expansion Theorem for Determinants
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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:
or:
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 $c$.
Pages in category "Expansion Theorem for Determinants"
The following 3 pages are in this category, out of 3 total.