Sylvester's Determinant Theorem
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High Quality Content by WIKIPEDIA articles! In algebra, the determinant is a special number associated with any square matrix. The fundamental geometric meaning of a determinant is a scale factor for measure when the matrix is regarded as a linear transformation. Thus a 2 × 2 matrix with determinant 2 when applied to a set of points with finite area will transform those points into a set with twice the area. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra. A matrix is invertible if and only if its determinant…mehr

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High Quality Content by WIKIPEDIA articles! In algebra, the determinant is a special number associated with any square matrix. The fundamental geometric meaning of a determinant is a scale factor for measure when the matrix is regarded as a linear transformation. Thus a 2 × 2 matrix with determinant 2 when applied to a set of points with finite area will transform those points into a set with twice the area. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra. A matrix is invertible if and only if its determinant is non-zero. The determinant of a matrix A, is denoted det(A), or without parentheses: det A. An alternative notation, used in the case where the matrix entries are written out in full, is to denote the determinant of a matrix by surrounding the matrix entries by vertical bars instead of the usual brackets or parentheses.