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The inverse of a nonsingular square matrix of dimension may be found by appending the identity matrix to the right of to form the dimensional augmented matrix (|). Applying elementary row operations to transform the left-hand n × n {\displaystyle n\times n} block to the identity matrix I {\displaystyle \mathbf {I} } , the right-hand n × n ...
By performing row operations, one can check that the reduced row echelon form of this augmented matrix is [|] = []. One can think of each row operation as the left product by an elementary matrix . Denoting by B the product of these elementary matrices, we showed, on the left, that BA = I , and therefore, B = A −1 .
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.
The inverse of a matrix is its adjugate matrix divided by its determinant: Augmented matrix: Matrix whose rows are concatenations of the rows of two smaller matrices: Used for performing the same row operations on two matrices Bézout matrix: Square matrix whose determinant is the resultant of two polynomials: See also Sylvester matrix ...
The row space is defined similarly. The row space and the column space of a matrix A are sometimes denoted as C(A T) and C(A) respectively. [2] This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces and respectively. [3]
The Gaussian-elimination consists of performing elementary row operations on the augmented matrix [] = [] for putting it in reduced row echelon form. These row operations do not change the set of solutions of the system of equations.
This demonstrates also, that operations on rows (e.g. pivoting) are equivalent to those on columns of a transposed matrix, and in general choice of row or column algorithm offers no advantage. In the lower triangular matrix all elements above the main diagonal are zero, in the upper triangular matrix, all the elements below the diagonal are zero.
Using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. The technique requires that all vectors be augmented with a "1" at the end, and all matrices be augmented with an extra row of zeros at the bottom, an extra column—the translation vector ...