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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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