DEFINITION:
If A and B are m × n matrices, then the
sum A+B is the m×n matrix whose entries are the sums of the corresponding
entries of A and B.
EXAMPLE:
1 −2 −1
0 1 2
1 −1 1
+
=
−2 3 −2
−2 −4 −1
−4 −1 −3
REMARK: We can add matrices only
of the same size.
EXAMPLE:


1 2
1 2
+  1 0  =???
3 4
3 4
DEFINITION:
If r is a scalar and A is a matrix, then
the scalar multiple rA is the matrix whose
entries are r times the corresponding
entries in A.
EXAMPLE:
1 2 −3
−2 −4 6
(−2)
=
−1 0 −2
2 0 4
PROPERTIES:
Let A, B, and C be matrices of the same
size, and let r and s be scalars. Then
(a) A + B = B + A
(b) (A + B) + C = A + (B + C)
(c) r(A + B) = rA + rB
(d) (r + s)A = rA + sA
(e) r(sA) = (rs)A
PROBLEM:
Consider the following matrices:


1 0
3 −2
1 2 3


.
A=
,B = 2 1 ,C =
2 5
2 1 4
3 2
If possible, compute:
(a) AB
(b) AC + B 2
(c) AB + C 2
SOLUTION:
We have:
(a) AB =


1 0
14 8
1 2 3 

2 1 =
.
16 9
2 1 4
3 2
(b) Impossible.
(c) AB + C 2
3 −2
3 −2
14 8
+
=
2 5
2 5
16 9
=
=
5 −16
14 8
+
16 21
16 9
19 −8
.
32 30
PROPERTIES:
Let A be an m × n matrix, and let B
and C have sizes for which the indicated
sums and products are defined. Then
(a) A(BC) = (AB)C
(b) A(B + C) = AB + AC
(c) (B + C)A = BA + CA
(d) r(AB) = (rA)B = A(rB)
WARNING
1. In general, AB 6= BA.
EXAMPLE:
1 2
1 1
Let A =
and B =
. Then
0 1
2 0
AB =
BA =
1 2
0 1
1 1
2 0
1 1
2 0
1 2
0 1
=
So,
AB 6= BA.
=
5 1
2 0
1 3
2 4
WARNING
2. If AB = AC, then it is not true in
general that B = C.
EXAMPLE: Let
1 0
0 0
0 0
A=
,B =
, and C =
.
0 0
0 1
0 2
Then
AB =
AC =
1 0
0 0
1 0
0 0
0 0
0 1
0 0
0 2
=
=
0 0
0 0
0 0
0 0
So,
AB = AC,
but
B 6= C.
WARNING
3. If AB = 0, then it is not true in
general that A = 0 or B = 0.
EXAMPLE:
1 0
0 0
and B =
. Then
Let A =
0 0
0 1
AB =
1 0
0 0
0 0
0 1
=
0 0
0 0
So,
AB = 0,
but
A 6= 0
and
B 6= 0.
THE TRANSPOSE OF A MATRIX
DEFINITION:
Let A be an m × n matrix. The transpose of A is the n × m matrix, denoted
by AT , whose columns are formed from
the corresponding rows of A.
EXAMPLE:
1 2
A=
3 4


−3 1
B =  4 7
8 −5
C=
1 2 3 4
5 6 7 8
AT =
1 3
2 4
−3 4 8
1 7 −5


1 5
2 6
T

C =
3 7
4 8
BT =
PROPERTIES:
Let A and B denote matrices whose sizes
are appropriate for the following sums
and products. Then
(a) (AT )T = A
(b) (A + B)T = AT + B T
(c) (rA)T = rAT for any scalar r
(d) (AB)T = B T AT
THE INVERSE OF A MATRIX
DEFINITION:
The identity matrix I is the n×n matrix
of the form


1 0 0 ... 0 0
0 1 0 ... 0 0


0 0 1 ... 0 0

I=
 ............ 


0 0 0 ... 1 0
0 0 0 ... 0 1
MAIN PROPERTY:
AI = IA = A
DEFINITION:
An n × n matrix A is said to be invertible if there is an n × n matrix C such
that
CA = I
and
AC = I.
In this case, C is an inverse of A and is
denoted by A−1. So,
A−1A = I
and
AA−1 = I.
EXAMPLE:
2 5
−7 −5
−1
Let A =
. Then A
=
.
−3 −7
3 2
In fact, we have
2 5
−7 −5
1 0
−1
AA
=
=
−3 −7
3 2
0 1
and
A−1A =
−7 −5
3 2
2 5
−3 −7
=
1 0
.
0 1
THEOREM:
a b
Let A =
. If ad − bc 6= 0, then A
c d
is invertible and
1
d −b
−1
A
=
.
−c
a
ad − bc
If ad − bc = 0, then A is not invertible.
PROBLEM:
Solve the following system of equations:
(
x1 − 2x2 = 0
x1 + 4x2 = 6
SOLUTION:
We have:
Ax̄ = B ⇒ x̄ = A−1B,
therefore
x̄ =
x1
x2
=
−1 1 −2
0
1 4
6
1
4 2
0
=
6
6 −1 1
1 12
=
6 6
2
=
.
1