The determinant is a scalar function defined on square matrices that serves as a test for invertibility, vanishing if and only if the matrix is singular (non-invertible). This concept operates within linear algebra based on three foundational axioms: mapping the identity to one, reversing sign upon row exchange, and exhibiting multilinear behavior with respect to individual rows. These properties allow for the derivation of a general formula where the determinant equals the product of diagonal entries in triangular form scaled by (-1) raised to the power of necessary row exchanges.
The determinant is a scalar function defined on square matrices that serves as a test for invertibility, vanishing if and only if the matrix is singular (non-invertible). This concept operates within…