Advanced book on Mathematics Olympiad

2.3 Linear Algebra 63

2.3.2 Determinants............................................

The determinant of ann×nmatrixA= (aij)i,j, denoted by detAor|aij|, is the
volume taken with sign of then-dimensional parallelepiped determined by the row (or
column) vectors ofA. Formally, the determinant can be introduced as follows. Let
e 1 =( 1 , 0 ,..., 0 ),e 2 =( 0 , 1 ,..., 0 ),..., en=( 0 , 0 ,..., 1 )be the canonical basis
ofRn. The exterior algebra ofRnis the vector space spanned by products of the form
ei 1 ∧ei 2 ∧···∧eik, where the multiplication∧is distributive with respect to sums and is
subject to the noncommutativity ruleei∧ej=−ej∧eifor alli, j(which then implies
ei∧ei =0, for alli). If the row vectors of the matrixAarer 1 ,r 2 ,...,rn, then the
determinant is defined by the equality

r 1 ∧r 2 ∧···∧rn=(detA)e 1 ∧e 2 ∧···∧en.

The explicit formula is

detA=

∑

σ

sign(σ )a 1 σ( 1 )a 2 σ( 2 )···anσ (n),

with the sum taken over all permutationsσof{ 1 , 2 ,...,n}.
To compute the determinant of a matrix, one applies repeatedly the row operation
that adds to one row a multiple of another until the matrix either becomes diagonal or
has a row of zeros. In the first case this transforms the parallelepiped determined by the
row vectors into a right parallelepiped in standard position without changing its volume,
as suggested in Figure 13.

Figure 13

But it is not our purpose to teach the basics. We insist only on nonstandard tricks and
methods. A famous example is the computation of the Vandermonde determinant.

Example.Letx 1 ,x 2 ,...,xnbe arbitrary numbers(n≥ 1 ). Compute the determinant
∣∣
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∣

11 ··· 1

x 1 x 2 ··· xn

..

.

..

.

... ..

.

xn 1 −^1 x 2 n−^1 ···xnn−^1

∣∣

∣∣

∣∣

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∣

.