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Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the n × n matrices that has the four following properties: The determinant of the identity matrix is 1. The exchange of two rows multiplies the determinant by −1.
The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then
It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis ).
Thus the only alternating multilinear functions with () = are restricted to the function defined by the Leibniz formula, and it in fact also has these three properties. Hence the determinant can be defined as the only function det : M n ( K ) → K {\displaystyle \det :M_{n}(\mathbb {K} )\rightarrow \mathbb {K} } with these three properties.
However, in the case of a ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse ...
For more properties and a generalization of the partial trace, see traced monoidal categories. If A is a general associative algebra over a field k , then a trace on A is often defined to be any map tr : A ↦ k which vanishes on commutators; tr([ a , b ]) = 0 for all a , b ∈ A .