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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.
For example, in the following sequence of row operations (where two elementary operations on different rows are done at the first and third steps), the third and fourth matrices are the ones in row echelon form, and the final matrix is the unique reduced row echelon form.
The reduced form of the system is: = + = +, with vector of reduced form errors that each depends on all structural errors, where the matrix A must be nonsingular for the reduced form to exist and be unique. Again, each endogenous variable depends on potentially each exogenous variable.
In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively. Often the aim of reduction is to transform a matrix into its "row-reduced echelon form" or "row-echelon form"; this is the goal of Gaussian elimination.
LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.
Any matrix can be reduced by elementary row operations to a matrix in reduced row echelon form. Two matrices in reduced row echelon form have the same row space if and only if they are equal. This line of reasoning also proves that every matrix is row equivalent to a unique matrix with reduced row echelon form.
The last matrix is in reduced row echelon form, and represents the system x = −15, y = 8, z = 2. A comparison with the example in the previous section on the algebraic elimination of variables shows that these two methods are in fact the same; the difference lies in how the computations are written down.
If the matrix is further simplified to reduced row echelon form, then the resulting basis is uniquely determined by the row space. It is sometimes convenient to find a basis for the row space from among the rows of the original matrix instead (for example, this result is useful in giving an elementary proof that the determinantal rank of a ...