Download Row Echelon Form

Math 311: Topics in Applied Math 1
1: Matrices & Systems of Equations
1.2: Row Echelon Form


1 4 0 2
(b)  0 0 1 3 
0 0 0 1


1 −3 0
2
0 1 −2 
(c)  0
0
0 0
0
1 2 0 1 5
(d)
0 0 1 3 4


1 5 −2 0 3
 0 0
0 1 6 

(e) 
 0 0
0 0 0 
0 0
0 0 0


0 1 0
2
(f)  0 0 1 −1 
0 0 0
0
Summary
• A matrix is in row echelon form provided
– The first nonzero entry occuring in each
nonzero row is 1.
– If row k is not all zeros, the number of leading
zero entries in row k + 1 is greater than that of
row k.
– If there are rows of all zeros, they lie below
rows having nonzero entries.
• Gaussian elimination: The process of reducing an
augmented system matrix into row echelon form via Solution
elementary row operations.

−2
(a) The unique solution is x =  5 .
3
• An overdetermined system has more equations
than unknowns (m > n for the m × n coefficient
matrix). It is usually (but not always) inconsistent.

(b) There are no solutions since the last row implies
0 = 1, which is false.
• An underdetermined system has fewer equations
than unknowns (m < n for the m × n coefficient
matrix). It is usually (but not always) consistent
with infinitely many solutions.
(c) There are infinitely many solutions of the form


3t + 2
 , t ∈ R.
x= t
−2
• A matrix is in reduced row echelon form (rref)
provided
(d) There are infinitely many solutions of the form


5 − 2s − t


s

x=
 4 − 3t  , s,t ∈ R.
t
– It is in row echelon form.
– The first nonzero entry in each row is the only
nonzero entry in its column.
• Gauss-Jordan reduction: The process of reducing
an augmented system matrix into reduced row
echelon form via elementary row operations.
(e) There are infinitely many solutions of the form


3 − 5s + 2t


s
 , s,t ∈ R.
x=
• A homogeneous linear system is one for which the


t
constants to the right of the equal signs are all zeros.
6
There is always a trivial solution (all variables equal
to zero). If such a system is underdetermined, it has (f) There are infinitely many solutions of the form


infinitely many solutions.
t
x =  2  , t ∈ R.
−1
Examples
24/3
24/7
The augmented matrices that follow are in reduced row
echelon form. In each case, find the solution set of the
corresponding linear system.


1 0 0 −2
5 
(a)  0 1 0
0 0 1
3
Give a geometric explanation of why a homogeneous
linear system consisting of two equations in three
unknowns must have infinitely many solutions. What are
the possible numbers of solutions of a nonhomogeneous
2 × 3 linear system? Give a geometric explanation of
your answer.
1
Solution
• A homogeneous linear system consisting of two
equations in three unknowns represents two planes
in space which contain the origin, since (0, 0, 0) is a
solution because the system is homogeneous. The
planes intersect in a straight line or are coincident
(i.e., the two equations represent the same plane).
• A nonhomogeneous linear system consisting of two
equations in three unknowns represents two planes
in space. There are three possibilities.
– The planes are parallel and do not intersect. In
this case the system has no solutions.
– The planes intersect in a straight line. The
system has infinitely many solutions
represented by one parameter.
– The planes are coincident (i.e., the two
equations actually represent the same plane).
The system has infinitely many solutions
represented by two parameters.
26/17
Let (c1 , c2 ) be a solution of the 2 × 2 system
a11 x1 + a12 x2 = 0
a21 x1 + a22 x2 = 0
Show that, for any real number α, the ordered pair
(αc1 , αc2 ) is also a solution.
Solution
Substituting (x1 , x2 ) = (αc1 , αc2 ) into the left-hand sides
of the equations gives
a11 (αc1 ) + a12 (αc2 ) = α (a11 c1 + a12 c2 ) = α (0) = 0
a21 (αc1 ) + a22 (αc2 ) = α (a21 c1 + a22 c2 ) = α (0) = 0
since (c1 , c2 ) was a given solution. Therefore, (αc1 , αc2 )
is also a solution.
2