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:

    Right and proof are two crutches for everything bent and crooked that limps along.
    Franz Grillparzer (1791–1872)

    Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.
    Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.
    Andrew Michael Ramsay (1686–1743)