Section 10.5 Matrix Algebra
In this section, we define matrix operations such as addition, subtraction, and multiplication, and
we give a procedure for finding the inverse of intertible matrices.
A
matrix
is a rectangular array of numbers. A matrix that has
m
rows
and
n
columns
is said to
be an
m
×
n
matrix, and the
m
×
n
matrix is said to be of
order
m
×
n
. The element
a
ij
is in row
i
and column
j
of the matrix (
a
ij
).
(
a
ij
) =
a
11
a
12
· · ·
a
1
j
· · ·
a
1
n
a
21
a
22
· · ·
a
2
j
· · ·
a
2
n
· · ·
a
i
1
a
i
2
· · ·
a
ij
· · ·
a
in
· · ·
a
m
1
a
m
2
· · ·
a
mj
· · ·
a
mn
Examples of matrices.
1
2

1
3

2
2
1
2

1
3
2
4
1

1
0
2
1
2
3
4
17
Matrix Equality.
Two matrices
A
= (
a
ij
) and
B
= (
b
ij
) are
equal
iff
a
ij
=
b
ij
for all
i, j
. Note: Two matrices
are equal iff corresponding entries are identical.
Example.
1
2
3
4
=
1
2
3
4
2
1
3
4
6
=
1
2
3
4
Adding and Subtracting Matrices
Only matrices of the same order can be added and subtracted.
Matrices are added and sub
tracted term by term, i.e., if
A
= (
a
ij
) and
B
= (
b
ij
), then
A
+
B
= (
a
ij
+
b
ij
) and
A

B
= (
a
ij

b
ij
).
Example.
If
A
=
1
1

2
3
and
B
=
2
0

4
7
, then
A
+
B
=
3
1

6
10
A

B
=

1
1
2

4
Let
α
be a scalar and let
A
= (
a
ij
) be a matrix.
Then
scalar multiplication
is defined by
αA
= (
αa
ij
).
Example.
Let
α
= 2 and
A
=
1
2

2
3
. Then
αA
= 2
1
2

2
3
=
2
4

4
6
Some Properties of matrix addition
1
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A
+
B
=
B
+
A
A
+ (
B
+
C
) = (
A
+
B
) +
C
(
k
+
h
)
A
=
kA
+
hA
k
(
A
+
B
) =
kA
+
kB
Multiplication of Matrices
The 1
×
n
matrix
R
= [
r
1
, r
2
,
· · ·
, r
n
] is called a
row vector
. The
m
×
1 matrix
C
=
c
1
c
2
.
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