Editing Determinant (section)

===Characterization of the determinant===
The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an <math>n \times n</math> matrix ''A'' as being composed of its <math>n</math> columns, so denoted as
:<math>A = \big ( a_1, \dots, a_n \big ),</math>
where the [[column vector]] <math>a_i</math> (for each ''i'') is composed of the entries of the matrix in the ''i''-th column.

# <li value="A"> <math>\det\left(I\right) = 1</math>, where <math>I</math> is an [[identity matrix]].
# <li value="B"> The determinant is ''[[multilinear map|multilinear]]'': if the ''j''th column of a matrix <math>A</math> is written as a [[linear combination]] <math>a_j = r \cdot v + w</math> of two [[column vector]]s ''v'' and ''w'' and a number ''r'', then the determinant of ''A'' is expressible as a similar linear combination:
#: <math>\begin{align}|A|
  &= \big | a_1, \dots, a_{j-1}, r \cdot v + w, a_{j+1}, \dots, a_n | \\
  &= r \cdot | a_1, \dots, v, \dots a_n | + | a_1, \dots, w, \dots, a_n |
\end{align}</math>
# <li value="C">The determinant is ''[[alternating form|alternating]]'': whenever two columns of a matrix are identical, its determinant is 0:
#: <math>| a_1, \dots, v, \dots, v, \dots, a_n| = 0.</math>

If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any <math>n \times n</math> matrix ''A'' a number that satisfies these three properties.<ref>[[Serge Lang]], ''Linear Algebra'', 2nd Edition, Addison-Wesley, 1971, pp 173, 191.</ref> This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula.

To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a [[standard basis]] vector. These determinants are either 0 (by property&nbsp;9) or else ±1 (by properties 1 and&nbsp;12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.{{citation needed|date=May 2021}}