Axioms 1-3 and Properties I-III
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.
Adding this axiom on top of the previous two makes the determinant a unique function of a matrix.
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