Game Development Reference

In-Depth Information

(b) Let
E
be equal to the identity matrix after row
r
has been multiplied by a sca-

lar
a
. Then the entries of
E
are given by

δ

,

if

ir

≠

;

ij

E

=

(3.43)

ij

aδ

, f

ir

=

.

ij

The entries of the product
EM
are then given by

M

,

if

ir

≠

;

n

ij

(

)

EM

=

EM

=

(3.44)

ij

ik

kj

aM

, f

i

=

r

.

ij

k

=

1

Thus, row
r
of the matrix
M
has been multiplied by
a
.

(c) Let
E
be equal to the identity matrix after row
r
has been multiplied by a sca-

lar
a
and added to row
s
. Then the entries of
E
are given by

δ

,

if

is

≠

;

ij

E

=

(3.45)

ij

δ aδ

+

, f

is

=

.

ij

rj

The entries of the product
EM
are then given by

,

if

is

≠

;

n

M

ij

(

)

EM

=

EM

=

(3.46)

ij

ik

kj

M

+

aM

, f

i

=

s

.

k

=

1

ij

rj

Thus, row
r
of the matrix
M
has been multiplied by
a
and added to row
s
.

The matrix
E
that represents the result of an elementary row operation per-

formed on the identity matrix is called an
elementary
matrix. If we have to apply

k
elementary row operations to transform a matrix
M
into the identity matrix,

then

=

E
,

I

EE

(3.47)

kk

−

1

1

where the matrices

E E
are the elementary matrices corresponding to

the same
k
row operations applied to the identity matrix. This actually shows that

the product

,

,

,

12

k

E E
is equal to the inverse of
M
, and it is exactly what we

get when we apply the
k
row operations to the identity matrix concatenated to the

matrix
M
in Equation (3.34).

kk

−

1

1

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