advertisement

Row-reduced Echelon Form of a Matrix Math 2270-001 (Summer 2005) Example Solve Thursday, May 19, 2005 8 < x + y 2x + 2y : x y 2 1 4 A := 2 1 2 1 1 40 0 0 0 2 1 1 A0 := 4 0 0 0 0 So, the given system is equivalent to 8 <x + y : whose solution is: 1 + z w = 0 + z 3w = 0 + z + 3w = 0 3 1 1 1 2 1 35 1 1 3 3 1 1 R1 1 1 5 R2 2R1 2 2 R3 + R1 3 0 2 R1 + R2 1 1 5 R2 0 0 R3 + 2R2 2w = 0 z + w = 0 ; 0 = 0 (x; y; z; w) = ( s 2t; s; t; t); s; t 2 R Remarks The matrix A is not in rref, while A0 is. It should be clear by now that: Every matrix M can be transformed to a UNIQUE matrix in row-reduced echelon form via row operations. This matrix is denoted by rref(M ). Note that rref(M ) should be regarded as some kind of a \standard simplication" of M . Denition Let M be a matrix. Then rank(M ) := the number of nonzero rows in rref(M ) Notice the interpretation: rank(M ) = the number of \independent" rows of the matrix M . Suppose M 2 Cmn. Then M can be regarded as the coecient matrix of a system of m linear equations in n unknowns. The \eective" number of equations of the that system is precisely rank(M ). rank(M ) is the number of non-redundant equations in that system. The number of parameters of the solution(s) of that system is precisely n rank(M ), namely, the number of unknowns minus the \eective" number of equations.