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10 2 0
01 10
00 0 0
.
(3.26)
Since this matrix has a row of zeros, we can assign an arbitrary value to the vari-
able corresponding to the third column since it does not contain a leading entry;
in this case we set za
=
. The first two rows then represent the equations
y +=
2
a
x
0
0
,
(3.27)
−=
so the solution to the system can be written as
2
1
1
x
ya
z
   
   
=
   
   
.
(3.28)
   
Homogeneous linear systems always have at least one solution—the zero
vector. Nontrivial solutions exist only when the reduced form of the coefficient
matrix possesses at least one row of zeros.
3.3 Matrix Inverses
1
An nn
×
matrix M is invertible if there exists a matrix, which we denote by
M
,
M is called the inverse of M . Not
every matrix has an inverse, and those that do not are called singular . An exam-
ple of a singular matrix is any one that has a row or column consisting of all
zeros.
1
1
1
such that
MMMMI . The matrix
=
=
Theorem 3.9. A matrix possessing a row or column consisting entirely of zeros
is not invertible.
Proof. Suppose every entry in row r of an nn
×
matrix F is 0. For any nn
×
ma-
Σ k
(
)
trix G , the
rr entry of the product FG is given by
F G
. Since each of
=
1
rk
kr
(
)
the
r F is 0, the
rr entry of FG is 0. Since the inverse of F would need to pro-
(
)
duce a 1 in the
rr entry, F cannot have an inverse. A similar argument proves
the theorem for a matrix possessing a column of zeros.
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