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Characterization of the determinant

The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an n × n {\displaystyle n\times n} -matrix A as being composed of its n {\displaystyle n} columns, so denoted as

where the column vector a i {\displaystyle a_{i}} (for each i) is composed of the entries of the matrix in the i -th column.

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 n × n {\displaystyle n\times n} -matrix A a number that satisfies these three properties. 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 9) or else ±1 (by properties 1 and 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.