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If Gaussian elimination applied to a square matrix A produces a row echelon matrix B, let d be the product of the scalars by which the determinant has been multiplied, using the above rules. Then the determinant of A is the quotient by d of the product of the elements of the diagonal of B : det ( A ) = ∏ diag ( B ) d . {\displaystyle \det ...
A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it.
Two algorithms are suggested: [2] [3] Division-free algorithm — performs matrix reduction to triangular form without any division operation. Fraction-free algorithm — uses division to keep the intermediate entries smaller, but due to the Sylvester's Identity the transformation is still integer-preserving (the division has zero remainder).
A pivot position in a matrix, A, is a position in the matrix that corresponds to a row–leading 1 in the reduced row echelon form of A. Since the reduced row echelon form of A is unique, the pivot positions are uniquely determined and do not depend on whether or not row interchanges are performed in the reduction process.
In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is a triangular matrix, its determinant is the product of the entries of its diagonal. So, the determinant can be computed for almost free from the result of a Gaussian elimination.
The column space of a matrix A is the set of all linear combinations of the columns in A. If A = [a 1 ⋯ a n], then colsp(A) = span({a 1, ..., a n}). Given a matrix A, the action of the matrix A on a vector x returns a linear combination of the columns of A with the coordinates of x as coefficients; that is, the columns of the matrix generate ...
Therefore, the critical case occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint. The overdetermined case occurs when the system has been overconstrained — that is, when the equations outnumber the unknowns.
In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z.Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R n, the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only.