Axioms 1-3 and Properties I-III
Axioms
Axiom 1. Homogeneity. If denotes the matrix obtained from
by multiplying one of its rows by a number
then
This implies that for
and
are invertible simultaneously.
Axiom 2. If denotes the matrix obtained from
by adding one of the rows of
to another, then
.
We remember that adding one of the rows of to another corresponds to adding one equation of the system
to another and so this operation should not impact solvability of the system.
Axiom 3.
Adding this axiom on top of the previous two makes the determinant a unique function of a matrix.
Properties
Assuming that is of size
it is convenient to partition it into rows:
Property I. If one of the rows of is zero, then
Indeed, if then
and by Axiom 1
Property II. If we add to one of the rows of another row multiplied by some number
the determinant does not change.
Proof. If there is nothing to prove. Suppose
and we want to add
to
This result can be achieved in three steps.
a) Multiply by
By Axiom 1,
gets multiplied by
b) In the new matrix, add row to
By Axiom 2, this does not change the determinant.
c) In the resulting matrix, divide row numbered by
The determinant gets divided by
The determinant of the very first matrix will be the same as that of the very last matrix, while the th row of the last matrix will be
Property III. If rows of are linearly dependent, then
Proof. Suppose rows of are linearly dependent. Then one of the rows can be expressed as a linear combination of the others. Suppose, for instance, that
Multiply the
th row by
and add the result to the first row, for
Thereby we make the first row equal to
while maintaining the determinant the same by Property II. Then by Property I
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