Math 311 Lecture 5
RECALL. A matrix is in reduced row echelon form (rref )
iff
the all-zero rows are at the bottom,
the lead entry of a nonzero row is 1 and all other
entries in its column are 0,
no nonzero terms occur in the area below and left of a
lead variable.
CExamples of reduced row echelon matrices.
105
015
000
000
,
01006
00105
00014
,
1−400
0010
0001
0000
RECALL. The elementary row (or just row) operations are:
Interchange two rows.
Multiply a row by a nonzero constant.
Add to a given row, a multiple of another row.
DEFINITION. Two matrices are row equivalent iff you can
get from one to the other by a sequence of row
operations.
THEOREM. Using row operations, every matrix A can be
reduced to a matrix rref(A) in reduced row echelon
form.
PROOF. Repeatedly applying row operations in the
following order will convert any matrix to rref.
Pick an unprocessed row with a leftmost nonzero
entry.
Make the entry 1.
Move the row just above all other unprocessed rows.
Make all other entries in the lead variable column 0.
CConvert to rref with the above process.
0a2a-1 3a
57 911
3333
100-2
0103
0010
DEFINITION. A system of linear equations is homogeneous
iff all right-hand constants are 0 iff its matrix form is
AX = 0. Every homogeneous system has the trivial
solution X = 0. A solution with one or more nonzero
values is nontrivial.
Cis a homogeneous system.
x+y−w=0
−2z+4w=0
x = y = z = w = 0 is the trivial solution.
The rref form is:
x+y−w=0
z−2w=0
Writing the lead variables (x and z here) in terms of the
remaining arbitrary variables gets the general solution.
The general solution is: .
x
=−
y+w,z=2w,y,warbitrar
y
Picking a nonzero value for some arbitrary variable, say
y = 1, w = 0, gives a nontrivial solution:
x = -1, y = 1, z = 0, w = 0.
The general solution can also be written parametrically:
x = -t + s, z = 2s, y = t, w = s,
where t and s are arbitrary parameters.
A system with k equations can have at most k lead
variables, the rest will be arbitrary. Hence
THEOREM. (a) A homogeneous system with more
unknowns than equations has at least one arbitrary
variable.
(b) If a system has an arbitrary variable, giving it a
nonzero value gives a nontrivial solution.
(c) If a homogeneous system has no arbitrary variables,
then it has no nontrivial solutions.
CFind
X
=
x
y
if A=
01
10
and AX =X
iff iff iff
A
X=X
A
X=IX (A−I)X=0
−11
1−1
x
y
=
0
0
Solving gives x = y, y arbitrary. Hence
, y arbitrary, is the general solution.
X
=
y
y
How many solutions are there for 2x = 4? 0x = 0? 0x = 2?
THEOREM. ax = b has a unique solution if a 0, infinitely
many solutions if a = b = 0, no solution if a = 0 and b
0.
C x+2y−2z=4
−y+5z=2
x+y+(a2−13)z=a+2
Find all a such that there is
(a) a unique solution, (b) infinitely many solutions,
(c) no solution.
xyz
é
rref xyz
12-24 1088
0-15 2 0 1-5-2
11
a2-13 a+2 0 0 a2-16 a-4
The last equation is (a2−16)z=a−4
(a) unique: a 4 (b) many: a = 4 (c) none: a = -4
