Laplace Expansion - Proof

Proof

Suppose is an n × n matrix and For clarity we also label the entries of that compose its minor matrix as

for

Consider the terms in the expansion of that have as a factor. Each has the form

\sgn \tau\,b_{1,\tau(1)} \cdots b_{i,j} \cdots b_{n,\tau(n)} = \sgn \tau\,b_{ij} a_{1,\sigma(1)} \cdots a_{n-1,\sigma(n-1)}

for some permutation τ ∈ Sn with, and a unique and evidently related permutation which selects the same minor entries as Similarly each choice of determines a corresponding i.e. the correspondence is a bijection between and The permutation can be derived from as follows.

Define by for and . Then and

Since the two cycles can be written respectively as and transpositions,

And since the map is bijective,

= \sum_{\sigma \in S_{n-1}} (-1)^{i+j}\sgn\sigma\, b_{ij} a_{1,\sigma(1)} \cdots a_{n-1,\sigma(n-1)}

from which the result follows.

Read more about this topic:  Laplace Expansion

Famous quotes containing the word proof:

    “If any doubt has arisen as to me, my country [Virginia] will have my political creed in the form of a “Declaration &c.” which I was lately directed to draw. This will give decisive proof that my own sentiment concurred with the vote they instructed us to give.”
    —Thomas Jefferson (1743–1826)

    “a meek humble Man of modest sense,
    Who preaching peace does practice continence;
    Whose pious life’s a proof he does believe,
    Mysterious truths, which no Man can conceive.”
    —John Wilmot, 2d Earl Of Rochester (1647–1680)

    “The insatiable thirst for everything which lies beyond, and which life reveals, is the most living proof of our immortality.”
    —Charles Baudelaire (1821–1867)