Matrices
For grade 1, undergraduate students
Abstract. Matrix theory plays a very important role for solving linear
equations, representing some linear transformations such as T: n→ n ,
and applying in many other fields.
Keywords. Matrix; the operations of matrix
.Some notations
Definition. A rectangular array of numbers composed of m rows and n columns
 a11

a
A   21


 am1
a12
a22
am 2
... a1n 

... a2 n 


... amn 
is called an m n matrix (read m by n matrix1). We also say that the matrix A is of,
or has, size m n .
Remark. If there is possibility of confusing entries from two adjacent columns as
a product we will insert commas between the entries of a given row to carefully
distinguish which entry belongs to which column.
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 ) 1 im
1  j n
for A, or simply
A  (aij )
when m and n are understood from context.
Remark. The order of the subscripts is important; the first subscript denotes the
row and the second subscript the column to which an entry belongs.
Just as with vectors in
That is:
n
, two matrices are equal iff they have the same entries.
Definition. If A  (aij ) and
B  (bij ) are
m n matrices, then A  B iff
aij  bij for i=1,2…, m and
j=1,…,n.
Our study of linear transformations suggests the following definitions.
Definition. If A  (aij ) and
A  B , is the matrix C  (cij ) ,
B  (bij ) are two m n matrices, their sum,
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 C  (cij ) where cij  raij , i=1,2…, m and j=1,…,n.
The following result is a routine verification of definitions:
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 n=n. Introduce the m n matrix
Ers  (eij ) by the requirement
0 if i  r , j  s ,
eij  
 1 if i  r , j  s .
For example the 6  4 matrix E32 is
0 0 0 0


0 0 0 0
0 1 0 0
E32  
.
0 0 0 0
0 0 0 0


0 0 0 0
It is then a routine verification to prove:
Proposition 2. The vectors
Therefore dim M mn  mn .
EXAMPLE 1.
Ers | r  1, 2,..., m, s  1, 2,..., n form a basis for
M mn .
1
4   0  1 3
 3

 

0  1   2  9 4
 2
 2  1 0   7 6 1

 

1 1 4
 3 0

 2 2
0 9  1
 2  7  1 6 0

3 
 
4

1 
 
3
4
5
0 7

9 3
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 
EXAMPLE 3.
4 
 3 1   12

 

4  7 4    28 16 
 6 4   24 16 

 

Definition. If A  (aij ) is an m n matrix and B  (bij ) is an n  p matrix, their
matrix product A  B is the m  p matrix AB  (cij ) , where
n
ci j   a i kb
k 1
k j
where i  1,..., m, j  1,..., p.
Thus the entry of the i-th row and j-th column of the product. A.B is obtained by
taking the i-th row
ai1 , ai 2. ., . ,ai n
of the matrix A and the j-th column of the matrix B
b1 j ,
b2 j ,
b nj ,
multiplying the corresponding entries together and adding the resulting products, i.e.
ai1 b1 j  ai 2 b2 
j . . .  ai k b 
k j. . . a i b
n ,n j
where
i  1, 2,..., m, j  1, 2,... p.
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 A  B can be defined without A  B
being defined.
Compute the matrix product
EXAMPLE 4.
(1
 4
 
3 )  5 .
6
 
2
Solution. Note the answer is a 1 1 matrix.
 4
 
(1 2 3)   5   (4  10  18)  (32).
6
 
Remark. Note that the product
 4
 
5 ( 1
6
 
2
3)
is not defined.
EXAMPLE 5.
Compute the matrix product
0

0
0

1
0
0
1

1
0
Answer
0 0 1


0 0 0
0 0 0


EXAMPLE 6.
Let
 1 2 3
A

 4 5 6
and
0

0
0

1
0
0
1

1 .
0
1

B  1
1

2
2
2
3 4

3 4
3 4
Calculate the product A  B
Solution. We have
1 2 3
 1 2 3 

 1 2 3
4
5
6


1 2 3
4

4
4 
2 4 6 ,
3 6 9 ,
 4 8 1 2
 1 2  3


1 8 ,  1 6 2 0
 4  5 6 8 1 0 1 2 , 1 2 1 5 
 6 1 2 1 8 2 4


 1 5 3 0 4 5 6 0
Remark.
24
Note that the product B  A is not defined.
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
A
0
0
3
 a n d B
3
 2
0

1
Compute the matrix products AB and BA.
Solution. We have
1
A B 
0
0

3
3

2
0  3
 
1 6
0
,  B A
3
3

2
0

1
1 0


 0 3
3

2
0

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
n
is defined and denoted by A
EXAMPLE 8. Let
0 0
A
.
1 0
Calculate A2
Solution. We have
0 0 0 0 0 0
A2  


.
1 0 1 0 0 0
Thus not only does matrix multiplication behave strangely in which it is not
commutative, it is also possible for the square of a matrix with nonzero entries to have
only zero entries.
The rules of matrix operations
Now, we may summarize the basic rules of matrix operations in the following
formulas: (assume that the indicated operations are defined, that is , that the sizes are
correct for the operations to make sense.)
(1)
A+B=B+A
(2)
A+(B+C)=A+(B+C)
(3)
r(A+B)=rA+rB
(4)
A+0=A
(5)
0A=0
(6)
A+(-1)A=0
(7)
(r+s)A=rA+sA
(8)
(A+B)·C=A·C+C·B
(9)
0·A=0=A·0
(10)
A·(B·C)=(A·B) ·C
( where0  (0ij )andoij  0)
In discussing matrices it is convenient to distinguish certain special types of matrices.
Special types of matrices
The identity matrix, the identity matrix of size n is the square n  n matrix denoted
by I, where
0
1 0


0
0 1
1 ifi  j ,
I  0 0 1
0   (iij ) where iij  


0 ifi  j.


0 0
 1 

For example, the identity matrices of size1,2,3and 4 are
1
 1 0 0 
 1 0 
 0
1 , 
, 0 1 0,
 0 1 
 0
 0 0 1 
0
0 0 0

1 0 0
0 1 0

0 0 1
The following important facts are easily verified
IB=B for any n  p matrix B,
AI=A for any m n matrix A.
Scalar matrices. A square matrix A=  aij  is called a scalar matrix iff A=rI for some
number r.
For example
 3 0 0
 1 0 0




 0 3 0   3 0 1 0 
 0 0 3
0 0 1




is a scalar matrix but
 3 0 0


 0 3 0
 0 0 3


is not scalar matrix.
The following formulas are easily checked
(1)
(aI)B=aB
for any n  p matrix B.
(2)
A(aI)=aA
for any m n matrix B.
For example
 3 0 0  1   3 
 1

   
 
 0 3 0  2    6   3  2 
 0 0 3  3   9 
 3

   
 
Diagonal matrices.
For any square matrix A= ( aij ) of size n,
a11 , a22 ,
the entries
, ann
are called the diagonal entries of A. For example, the diagonal entries of
3 2 1


 6 5 4
9 8 7


are 3,5,7. A square matrix is said to be a diagonal matrix iff its only nonzero entries
are on the diagonal. That is A= ( aij ) is a diagonal matrix iff ai  0 for i  j .
For example I and aI are diagonal matrices as it is
 3 0 0


 0 3 0 
 0 0 3


Remark. The diagonal entries themselves need not be nonzero . For example
 0 0 0


 0 1 0  and
 0 0 0


 0 0 0


 0 0 0
 0 0 0


are also diagonal matrices.
In general a diagonal matrix looks like
 a11

A



0 




ann 
a22
0
where the giant 0s mean that all other entries are zero. If A and B are diagonal
matrices of size n then so are AB and BA. Indeed if
 a11

A
 0

0 

 and
a nn 
 b11

B 
0

0 


b nn 
then
 a11b11

AB  


 0
a22b22
0 

 =BA


an bn 
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.
A triangular matrix A= ( aij ) where aii  0, i  1,
, n (that is ,all of whose
diagonal entries are 0) is said to be strictly triangular.
An example of a strictly triangular matrix is
 0 0 0


 1 0 0
 2 3 0


The Zero matrix.
The zero matrix of size m n is the m n matrix 0 all of those
entries are 0.
Idempotent matrices.
A square matrix A is said to be idempotent iff A  A .
2
There are lots of idempotent matrices. Here are a few examples
 1 1

,
0 0 
 1 0 0


 0 1 0
 0 0 0


as may be easily checked by explicit computation.
Nilpotent matrices. A square matrix A is said to be nilpotent iff there is an integer q such
Aq  0 .(The smallest such integer q is called the index of nilpotence of A).
3
For example if A is the matrix of the shift operator on R , that is
 1 0 0


A   0 1 0
 0 0 0


then
 0 0 0  0 0 0   0 0 0 


 

A   1 0 0  1 0 0    0 0 0 
 0 1 0  0 1 0   1 0 0 


 

2
and
 0 0 0  0 0 0   0 0 0 


 

A   0 0 0  1 0 0    0 0 0 
 1 0 0  0 1 0   0 0 0 


 

3
so the A is nilpotent of index 3.
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.
If A is nonsingular then the matrix B With AB=I=BA is called the inverse matrix of A and is
denoted by A1 .
It A is nonsingular then the matrix B with AB=I=BA is called the inverse matrix of A and is
denoted by A1
It is a theorem that we can prove that if there exists a matrix B such that
AB=I.
Then also
BA=I.
1
Thus to check that B= A we need only calculate one of the two products AB and BA and see if
they are I. For example if
 0 1 0


A   1 0 0
 0 0 1


then
 0 1 0


A1   1 0 0 
 0 0 1


for we have
 0 1 0  0 1 0  0 1 0 




AA   1 0 0  1 0 0  1 0 0   I
 0 0 1  0 0 1  0 0 1 




1
and therefore A= A .
An example of a matrix that is not invertible is
 0 0 0


1 0 0
 0 1 0


and more generally we have ;
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
Aq  0
Then
0  Aq B  Aq 1 AB  Aq 1I  Aq 1
so
Aq 1  0
We may then repeat the above trick to show
Aq  2  0
If we repeat this trick q  1 times we will get
A0
But then
I  AB  0B  0
which is impossible.
We may also show:
The only invertible idempotent matrix is I.
For if A is an idempotent matrix then
A A
2
implies
A  IA  BAA  BA2  BA  I .
So A=I as claimed.
Symmetric and skew-symmetric matrices.
A square matrix A=
 a  is said to symmetric
ij
iff aij  a ji for i, j..., n; it is said to be skew-symmetric iff aij  -a ji for 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 


 1 0 3  and
 -2 3 0 


 0 1


 1 0 
are skew-symmetric matrices. Notice that the matrix
 1 0 1


 0 0 0
 -1 1 3 


is not skew-symmetric because
a  1  1  a11
 
That is to say, if a matrix A= aij is skew-symmetric then the equations
a11  a11; a22  a22 ,...,ann  ann
certainly imply that a11  0; a22  0,...,ann  0 , that is skew-symmetric matrix has all its
diagonal entries equal to 0.
The skew-symmetric matrix
 0 1
A

 1 0 
is interesting because it is also nonsingular since
 0 1  0 1  1 0 


＝
.
 1 0  1 0   0 1 
Proposition 3
A 2  2 matrix
a b
A

c d
is nonsingular iff ad  bc  0.
A1 
If ad  bc  0, then
1  d b 


ad  bc  c a 
PROOF. Suppose that ad  bc  0.
B
,Let
1  d b 


ad  bc  c a 
Then
BA 
1  d b  a b 



ad  bc  c a  c d 

1  da  bc bd  bd 


ad  bc  ca  ac ca  ad 

0 
1  ad  bc


ad  bc 
ad  bc  0
1 0

I
0 0
and therefore A is nonsingular with
A1 
1  d b 


ad  bc  c a 
as claimed.
Suppose conversely that A is nonsingular, but that
contradiction. Let
ad  bc  0 . We will deduce a
 d b 
C 

 c a 
Then computing as above
0 
 ad  bc
CA  
  (ad  bc) I  0.
ad  bc 
 0
This gives the equation
C  CI  C ( AA1 )  (CA) A1  0 A1  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
I  AA1  0 A1  0
so
 1 0  0 0 



 0 1  0 0 
and hence 1=0, which is impossible.
SOME EXERCISES
1. Perform the following matrix multiplications
 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 



2. Which of the following matrices are nonsingular, idempotent, nilpotent, symmetric, or
skew-symmetric?
 1 1
 0 1
A
 F 

0 0 
 1 0 
 1 1
 1 0
B
 G 

 1 1 
 0 1
 1 1
 1 0
C 
 H 

 1 1 
 1 0 
 1 1
 4 0
D
 J 

 1 1
 0 2
 1 1
 0 0
E 
 K 

 1 1
 1 0
Find the inverse for those that are invertible.
 
 
3. If A  aij is a matrix we define the transpose of A to be the matrix A  aij
where
bij  aij . Find the transpose of each of the following matrices:
 1 2 3

 1 2 3
 4 5 6
 1 0 11 2



 0 1 0 3 4
 1 0 05 6



Show for any matrix A that
 A  =A.
t t
4. Let A be a square matrix. Show that A is symmetric iff A  A . A is skew-symmetric iff
t
A   At .
5. For any square matrix A show that A  At is symmetric and A  At is skew-symmetric.
6. Let A be an idempotent matrix. Show that I-A is also idempotent.
7. A square matrix A is said to commute with a matrix B iff AB=BA. When does a 3  3 matrix
A commute with the matrix Ers ?
8. Show that if a 3  3 matrix A commutes with every 3  3 matrix B then A is a scalar matrix.
(Hint: If A commutes with every matrix B it commutes with the 9 matrices Ers , r , s  1, 2,3. ).
9. Find all 2  2 matrices that commute with
 1 1

.
 0 1
10. Construct a 3  3 matrix A such that A  I .
3
11. Let A be a 3  3 matrix, D be the diagonal matrix
 d1 0

D   0 d2
0 0

(a)
(b)
0

0 .
d3 
Compute D:A
Compute A:D
12. Let A be 3  3 matrix. Compute E12  A, A  E12 , E21  A, A  E21 . What conclusion can
you obtain in general for Ers .A and A Ers ?
13. 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.)