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.