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Matrices
For grade 1, undergraduate students
Made by Department of Math. ,Anqing Teachers college
1 Some notations
Definition. A rectangular array of numbers
composed of m rows and n columns
 a11

a
21

A


 am1
a12
a22
am 2
... a1n 

... a2 n 


... amn 
is called an m n matrix (read m by n
matrix). We also say that the matrix A is of, or
has, size m n .
The elements
ai1 , ai 2
, ain
form the i-th row of A ,
and the elements
a1 j , a2 j ,..., a mj
form the j-th column of A. We will often
write
A  (aij ) mn
for A.
Definition. If
A  (aij ) and
B  (bij )
are m n matrices, then A  B
aij  bij
for i=1,2…, m and j=1,…,n.
iff
.Matrix opertions
Definition. If
A  (aij ) and
B  (bij )
are two m n matrices, their sum A+B , is the
matrix C  (c )
ij
, where
cij  aij  bij ,
i=1,2…, m , j=1,2…,n.
Definition. If A  (aij ) is an m n
matrix and
r is a number then rA, the scalar multiple of A
by r, is the matrix
where
C  (cij )
cij  raij ,
i=1,2…, m and j=1,…,n.
.Some properties
Proposition 1. The matrices of size m n form
a vector space under the operations of matrix
addition and scalar multiplication. We denote this
vector space by Mmn.
The dimension of the vector space Mmn is not
hard to compute. We take our lead from the
method we used to show that dim Rn=n.
Introduce the m nmatrix Ers  (eij )
by the requirement
0 if i  r , j  s,
eij  
 1 if i  r , j  s.
Proposition 2. The vectors
Ers | r  1, 2,..., m, s  1, 2,..., n
form a basis for Mmn . Therefore
dim M mn  mn
EXAMPLE 1.
 3 1

 2 0
 2 1

 3 0

  22
 2  7

4   0 1 3 
 

1   2 9 4 
0   7 6 1 
1 1
4 3  3 0 7
 

0  9 1  4    4 9 3  .
1  6 0  1   5 5 1 
EXAMPLE 2.
 1 4 6 0 1   0 1 1 4 7 



 2 0 1 7 9   1 2 3 7 9 
1  0 4  1 6  1 0  4 1  7 


 2  1 0  2 1  3 7  7 9  9 
1 5 5 4 8 


1

2

4
14
18


2 Matrix products
Definition. If A  (aij ) is an m n matrix
n p
and B  (bij ) is an
matrix, their
matrix product A  B
is the m  p matrix,
where
AB  (cij )
n
cij   aik bkj i  1,..., m, j  1,..., p.
k 1
Remark. Note that for the product of A
and B to be defined the number of
columns of A must be equal to the
number of rows of B. Thus the order in
which the product of A and B is taken is
very important, for AB can be defined
without AB being defined.
EXAMPLE 4. Compute the matrix
product
 4
 
(1 2 3)   5  .
6
 
Solution. Note the answer is a
matrix
 4
 
(1 2 3)   5   (4  10  18)  (32).
6
 
Remark. Note that the product
 4
 
 5   (1 2 3)
6
 
is not defined.
EXAMPLE 5. Compute the matrix product
0 1 1 0 1 1

 

0
0
1

0
0
1

 
.
0 0 0 0 0 0

 

Answer .
0 0 1


0 0 0
0 0 0


 Definition. A matrix A is said to be a square
matrix of size n iff it has n rows and n
columns (that is the number of rows equals
the number of columns equals n).
Remark. It is easy to see that if A and B
are square matrices of size n then the
products AB and BA are both defined.
However they may not be equal..
EXAMPLE 7. Let
1 0
 3 0
A
 and B  

 0 3
 2 1
Compute the matrix products AB and BA.
Solution. We have
1 0  3 0  3 0
 3 0 1 0  3 0
AB  


 , BA  



 0 3  2 1  6 3
 2 1  0 3  2 3
and so we see that AB
 BA.
Remark. As the preceding example
shows even if AB and BA are defined we
should not expect that AB=BA.
Notation. If A is a square matrix then AA
is defined and is denoted by A2.
Similarly,
A... A
n times
is defined and denoted by A .
n
EXAMPLE 8. Let
0 0
A
.
1 0
Calculate A2 .
Solution. We have
 0 0  0 0  0 0
A 



1 0 1 0 0 0
2
.The rules of matrix operations
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
A+B=B+A
A+(B+C)=A+(B+C)
r(A+B)=rA+rB
A+0=A
0A=0
A+(-1)A=0
(r+s)A=rA+sA
(A+B)·C=A·C+C·B
0·A=0=A·0
A·(B·C)=(A·B) ·C
3 Special types of matrices
Diagonal matrices.
 a11

A
 0

0 


a nn 
Triangular matrices. A square matrix A is
said to be lower triangular iff A= ( aij )
where aij  0 if j  i
For example
1 0 0 


0
2
0


 3  1 3 


is a lower triangular matrix.
The Zero matrix. The zero matrix is the
matrix 0 all of those entries are 0.
Idempotent matrices. A square matrix A
is said to be idempotent iff A2  A
Nilpotent matrices. A square matrix A is
said to be nilpotent iff there is an integer q
such Aq  0 .
Nonsingular matrices. A square matrix A
is said to be invertible or nonsingular iff
there exists a matrix B such that
AB=I and BA=I.
1
A
Denoted by .
For example if
then
0 1 0


A  1 0 0
0 0 1


0 1 0


A 1   1 0 0  .
0 0 1


A nilpotent matrix is not invertible. For
suppose that A is a nilpotent matrix that is
invertible. Let B be an inverse for A. Since A
is nilpotent there is an integer q such that
A  0.
q
Then
0 A B A
q
so
q 1
q 1
AB  A I  A
q 1
Aq 1  0.
If we repeat this trick q-1 times we
will get
A  0.
But then
I  AB  0 B  0,
which is impossible.
Symmetric and skew-symmetric matrices. A
square matrix A= aij is said to symmetric iff
 
aij  a ji
for it is said to be skew-symmetric iff for
aij  -a ji , i, j  1, 2,..., n.
For example
1 0 1


0
0
0

 and
1 1 3


0

1
2

3
1 2 3

4 5 6
5 7 8

6 8 9
are symmetric matrices, and
 1 1 2 
 0 1


 1 0 3  and  1 0 


 -2 3 0 


are skew-symmetric matrices.
Proposition 3 A
2 2
matrix
a b
A

c d 
is nonsingular iff ad  bc  0.
If ad  bc  0,
then
1  d b 
A 


ad  bc  c a 
1
PROOF. Suppose that ad
Let
1  d b 
B


ad  bc  c a 
Then
 bc  0.
1  d b  a b 
BA 




c
a
c
d
ad  bc 


1  da  bc bd  bd 



ad  bc  ca  ac ca  ad 
0 
1  ad  bc



0
ad

bc
ad  bc 

1 0

  I.
0 0
and therefore A is nonsingular with
1  d b 
A 

.
ad  bc  c a 
1
Suppose conversely that A is nonsingular,
but that
ad  bc  0 . We will deduce a
contradiction. Let
 d b 
C 
.
 c a 
Then computing as above
0 
 ad  bc
CA  
  (ad  bc) I  0.
ad  bc 
 0
This gives the equation
1
1
1
C  CI  C ( AA )  (CA) A  0 A  0.
Therefore
 d b  0 0 
C 

.
 c a  0 0 
So that
a  0, b  0, c  0, d  0.
But then A=0 also, so
1
1
I  AA  0 A  0.
So
 1 0  0 0 



0
1
0
0



and hence 1=0, which is impossible.
4 SOME EXERCISES
1.
Perform
the
multiplications
following
 0 1 0  1 0 



1
1
0
0
1


,
 0 0 2  1 0 



1 0


1 0 1 00 1 2 2
,




0 1 0 1 1 0 2 2


0
1


 0 0 0  1 2 3 



1
0
0
4
5
6


.
 0 0 0  7 8 9 



matrix
2. Which of the following matrices are
nonsingular,
idempotent,
nilpotent,
symmetric, or skew-symmetric?
 1 1
A
,
0 0 
 1 1
B
,
 1 1 
 1 1
C 
,
 1 1 
1 1
D
,
1 1
 0 1
F 
,
 1 0 
1 0
G 
,
0 1
 1 0
H 
,
 1 0 
4 0
J 
.
0 2
3. If A is an idempotent square matrix
show I-2A is invertible (Hint: Idempotent
correspond to projections. Interpret I-2A as a
reflection. Try the 2  2 case first. Then try to
generalize.)
Thanks!!!