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Determinants axiomatically

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The determinant function is unique.
Definition of multilinear function
The determinant function exists.
A determinant is a multilinear function
Adding a multiple of one row to another row does not change the determinant.
Switching two rows multiplies the determinant by -1.
Multiplying a row by a scalar multiplies the determinant by that scalar.
A matrix with two equal rows/columns has determinant 0
A matrix with a 0 row/column has determinant 0

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Multiplying a row by a scalar multiplies the determinant by that scalar.

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A determinant is a multilinear functionClose match
Adding a multiple of one row to another row does not change the determinant.Close match
Definition of the determinant in terms of the effect of elementary row operationsClose match
Determinants and operations on matricesClose match
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The determinant of a matrix can be computed as a cofactor expansion across any row.Close match
Determinants multiply: det(AB) = det(A)det(B)Close match
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A matrix and its transpose have the same determinant.Close match
The determinant of a matrix can be computed as a cofactor expansion down any column.Close match
The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form.Close match
Determinants axiomaticallyClose match
Definition of determinant of a matrix as a cofactor expansion across the first rowClose match
Matrix-scalar multiplication is commutativeClose match
Definition of matrix-scalar multiplicationClose match
Scalar multiplication of matrices is associative and distributiveClose match
A matrix with two equal rows/columns has determinant 0Close match
Switching two rows multiplies the determinant by -1.Close match
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The determinant of the matrix of a linear transformation is the factor by which the area/volume changes.Close match

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