Matrices | Engineering Mathematics - Engineering Mathematics PDF Download

 Page 1


Matrices
1.1 De?nition of a Matrix
De?nition 1.1.1 (Matrix) A rectangular array of numbers is called a matrix.
We shall mostly be concerned with matrices having real numbers as entries.
The horizontal arraysof a matrix are called itsrows and the vertical arraysare called itscolumns.
A matrix having m rows and n columns is said to have the orderm×n.
A matrix A of order m×n can be represented in the following form:
A =
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
,
where a
ij
is the entry at the intersection of the i
th
row andj
th
column.
In a more concise manner, we also denote the matrix A by [a
ij
] by suppressing its order.
Remark 1.1.2 Some books also use
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
to represent a matrix.
LetA =
"
1 3 7
4 5 6
#
. Then a
11
= 1, a
12
=3, a
13
= 7, a
21
= 4, a
22
= 5, and a
23
=6.
A matrix having only one column is called a column vector; and a matrix with only one row is
called a row vector.
Whenever a vector is used, it should be understood from the context whether it is
a row vector or a column vector.
De?nition 1.1.3 (Equality of two Matrices) Two matricesA = [a
ij
] andB =[b
ij
] having the same order
m×n are equal if a
ij
=b
ij
for each i =1,2,...,m and j =1,2,...,n.
In other words,two matricesaresaid to be equal if they havethe same orderand their corresponding
entries are equal.
9
Page 2


Matrices
1.1 De?nition of a Matrix
De?nition 1.1.1 (Matrix) A rectangular array of numbers is called a matrix.
We shall mostly be concerned with matrices having real numbers as entries.
The horizontal arraysof a matrix are called itsrows and the vertical arraysare called itscolumns.
A matrix having m rows and n columns is said to have the orderm×n.
A matrix A of order m×n can be represented in the following form:
A =
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
,
where a
ij
is the entry at the intersection of the i
th
row andj
th
column.
In a more concise manner, we also denote the matrix A by [a
ij
] by suppressing its order.
Remark 1.1.2 Some books also use
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
to represent a matrix.
LetA =
"
1 3 7
4 5 6
#
. Then a
11
= 1, a
12
=3, a
13
= 7, a
21
= 4, a
22
= 5, and a
23
=6.
A matrix having only one column is called a column vector; and a matrix with only one row is
called a row vector.
Whenever a vector is used, it should be understood from the context whether it is
a row vector or a column vector.
De?nition 1.1.3 (Equality of two Matrices) Two matricesA = [a
ij
] andB =[b
ij
] having the same order
m×n are equal if a
ij
=b
ij
for each i =1,2,...,m and j =1,2,...,n.
In other words,two matricesaresaid to be equal if they havethe same orderand their corresponding
entries are equal.
9
10 CHAPTER 1. MATRICES
Example 1.1.4 The linear system of equations 2x +3y = 5 and 3x+2y = 5 can be identi?ed with the
matrix
"
2 3 : 5
3 2 : 5
#
.
1.1.1 Special Matrices
De?nition 1.1.5 1. A matrix in which each entry is zero is called a zero-matrix, denoted by 0. For
example,
0
2×2
=
"
0 0
0 0
#
and 0
2×3
=
"
0 0 0
0 0 0
#
.
2. A matrix having the number of rows equal to the number of columns is called a square matrix. Thus,
its order is m×m (for some m) and is represented by m only.
3. In a square matrix, A = [a
ij
], of order n, the entries a
11
,a
22
,...,a
nn
are called the diagonal entries
and form the principal diagonal ofA.
4. A square matrix A = [a
ij
] is said to be a diagonal matrix if a
ij
= 0 for i 6= j. In other words, the
non-zero entries appear only on the principal diagonal. For example, the zero matrix 0
n
and
"
4 0
0 1
#
are a few diagonal matrices.
AdiagonalmatrixDofordernwiththediagonalentriesd
1
,d
2
,...,d
n
isdenotedbyD = diag(d
1
,...,d
n
).
If d
i
=d for all i =1,2,...,n then the diagonal matrix D is called a scalar matrix.
5. A square matrix A = [a
ij
] with a
ij
=
(
1 if i=j
0 if i6=j
is called the identity matrix, denoted byI
n
.
For example,I
2
=
"
1 0
0 1
#
, and I
3
=
?
?
?
1 0 0
0 1 0
0 0 1
?
?
?.
The subscript n is suppressed in case the order is clear from the context or if no confusion arises.
6. A square matrix A = [a
ij
] is said to be an upper triangular matrix if a
ij
=0 fori>j.
A square matrix A = [a
ij
] is said to be an lower triangular matrix if a
ij
= 0 for i<j.
A square matrix A is said to be triangular if it is an upper or a lower triangular matrix.
For example
?
?
?
2 1 4
0 3 -1
0 0 -2
?
?
? is an upper triangular matrix. An upper triangular matrix will be represented
by
?
?
?
?
?
?
a
11
a
12
··· a
1n
0 a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
0 0 ··· a
nn
?
?
?
?
?
?
.
1.2 Operations on Matrices
De?nition 1.2.1 (Transpose of a Matrix) The transpose of an m×n matrix A = [a
ij
] is de?ned as the
n×m matrix B =[b
ij
], with b
ij
=a
ji
for 1=i=m and 1=j=n. The transpose of A is denoted byA
t
.
Page 3


Matrices
1.1 De?nition of a Matrix
De?nition 1.1.1 (Matrix) A rectangular array of numbers is called a matrix.
We shall mostly be concerned with matrices having real numbers as entries.
The horizontal arraysof a matrix are called itsrows and the vertical arraysare called itscolumns.
A matrix having m rows and n columns is said to have the orderm×n.
A matrix A of order m×n can be represented in the following form:
A =
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
,
where a
ij
is the entry at the intersection of the i
th
row andj
th
column.
In a more concise manner, we also denote the matrix A by [a
ij
] by suppressing its order.
Remark 1.1.2 Some books also use
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
to represent a matrix.
LetA =
"
1 3 7
4 5 6
#
. Then a
11
= 1, a
12
=3, a
13
= 7, a
21
= 4, a
22
= 5, and a
23
=6.
A matrix having only one column is called a column vector; and a matrix with only one row is
called a row vector.
Whenever a vector is used, it should be understood from the context whether it is
a row vector or a column vector.
De?nition 1.1.3 (Equality of two Matrices) Two matricesA = [a
ij
] andB =[b
ij
] having the same order
m×n are equal if a
ij
=b
ij
for each i =1,2,...,m and j =1,2,...,n.
In other words,two matricesaresaid to be equal if they havethe same orderand their corresponding
entries are equal.
9
10 CHAPTER 1. MATRICES
Example 1.1.4 The linear system of equations 2x +3y = 5 and 3x+2y = 5 can be identi?ed with the
matrix
"
2 3 : 5
3 2 : 5
#
.
1.1.1 Special Matrices
De?nition 1.1.5 1. A matrix in which each entry is zero is called a zero-matrix, denoted by 0. For
example,
0
2×2
=
"
0 0
0 0
#
and 0
2×3
=
"
0 0 0
0 0 0
#
.
2. A matrix having the number of rows equal to the number of columns is called a square matrix. Thus,
its order is m×m (for some m) and is represented by m only.
3. In a square matrix, A = [a
ij
], of order n, the entries a
11
,a
22
,...,a
nn
are called the diagonal entries
and form the principal diagonal ofA.
4. A square matrix A = [a
ij
] is said to be a diagonal matrix if a
ij
= 0 for i 6= j. In other words, the
non-zero entries appear only on the principal diagonal. For example, the zero matrix 0
n
and
"
4 0
0 1
#
are a few diagonal matrices.
AdiagonalmatrixDofordernwiththediagonalentriesd
1
,d
2
,...,d
n
isdenotedbyD = diag(d
1
,...,d
n
).
If d
i
=d for all i =1,2,...,n then the diagonal matrix D is called a scalar matrix.
5. A square matrix A = [a
ij
] with a
ij
=
(
1 if i=j
0 if i6=j
is called the identity matrix, denoted byI
n
.
For example,I
2
=
"
1 0
0 1
#
, and I
3
=
?
?
?
1 0 0
0 1 0
0 0 1
?
?
?.
The subscript n is suppressed in case the order is clear from the context or if no confusion arises.
6. A square matrix A = [a
ij
] is said to be an upper triangular matrix if a
ij
=0 fori>j.
A square matrix A = [a
ij
] is said to be an lower triangular matrix if a
ij
= 0 for i<j.
A square matrix A is said to be triangular if it is an upper or a lower triangular matrix.
For example
?
?
?
2 1 4
0 3 -1
0 0 -2
?
?
? is an upper triangular matrix. An upper triangular matrix will be represented
by
?
?
?
?
?
?
a
11
a
12
··· a
1n
0 a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
0 0 ··· a
nn
?
?
?
?
?
?
.
1.2 Operations on Matrices
De?nition 1.2.1 (Transpose of a Matrix) The transpose of an m×n matrix A = [a
ij
] is de?ned as the
n×m matrix B =[b
ij
], with b
ij
=a
ji
for 1=i=m and 1=j=n. The transpose of A is denoted byA
t
.
1.2. OPERATIONS ON MATRICES 11
That is, by the transpose of an m×n matrix A, we mean a matrix of order n×m having the rows
of A as its columns and the columns of A as its rows.
For example, if A =
"
1 4 5
0 1 2
#
then A
t
=
?
?
?
1 0
4 1
5 2
?
?
?.
Thus, the transpose of a row vector is a column vector and vice-versa.
Theorem 1.2.2 For any matrix A, we have (A
t
)
t
=A.
Proof. Let A = [a
ij
], A
t
=[b
ij
] and (A
t
)
t
=[c
ij
]. Then, the de?nition of transpose gives
c
ij
=b
ji
=a
ij
for all i,j
and the result follows. 
De?nition 1.2.3 (Addition of Matrices) letA = [a
ij
] andB = [b
ij
] be are twom×n matrices. Then the
sum A+B is de?ned to be the matrix C =[c
ij
] with c
ij
=a
ij
+b
ij
.
Note that, we de?ne the sum of two matrices only when the order of the two matrices are same.
De?nition 1.2.4 (Multiplying a Scalar to a Matrix) Let A = [a
ij
] be an m×n matrix. Then for any
element k?R, we de?ne kA = [ka
ij
].
For example, if A =
"
1 4 5
0 1 2
#
and k =5, then 5A =
"
5 20 25
0 5 10
#
.
Theorem 1.2.5 Let A,B and C be matrices of order m×n, and let k,l?R. Then
1. A+B =B +A (commutativity).
2. (A+B)+C =A+(B +C) (associativity).
3. k(lA) = (kl)A.
4. (k+l)A =kA+lA.
Proof. Part 1.
Let A = [a
ij
] andB = [b
ij
]. Then
A+B = [a
ij
]+[b
ij
] = [a
ij
+b
ij
] =[b
ij
+a
ij
] = [b
ij
]+[a
ij
] =B +A
as real numbers commute.
The reader is required to prove the other parts as all the results follow from the properties of real
numbers. 
Exercise 1.2.6 1. Suppose A+B =A. Then show that B =0.
2. Suppose A+B =0. Then show that B = (-1)A = [-a
ij
].
De?nition 1.2.7 (Additive Inverse) Let A be an m×n matrix.
1. Then there exists a matrix B with A+B =0. This matrix B is called the additive inverse of A, and
is denoted by-A = (-1)A.
2. Also, for the matrix0
m×n
,A+0 =0+A=A. Hence, the matrix0
m×n
is called the additive identity.
Page 4


Matrices
1.1 De?nition of a Matrix
De?nition 1.1.1 (Matrix) A rectangular array of numbers is called a matrix.
We shall mostly be concerned with matrices having real numbers as entries.
The horizontal arraysof a matrix are called itsrows and the vertical arraysare called itscolumns.
A matrix having m rows and n columns is said to have the orderm×n.
A matrix A of order m×n can be represented in the following form:
A =
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
,
where a
ij
is the entry at the intersection of the i
th
row andj
th
column.
In a more concise manner, we also denote the matrix A by [a
ij
] by suppressing its order.
Remark 1.1.2 Some books also use
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
to represent a matrix.
LetA =
"
1 3 7
4 5 6
#
. Then a
11
= 1, a
12
=3, a
13
= 7, a
21
= 4, a
22
= 5, and a
23
=6.
A matrix having only one column is called a column vector; and a matrix with only one row is
called a row vector.
Whenever a vector is used, it should be understood from the context whether it is
a row vector or a column vector.
De?nition 1.1.3 (Equality of two Matrices) Two matricesA = [a
ij
] andB =[b
ij
] having the same order
m×n are equal if a
ij
=b
ij
for each i =1,2,...,m and j =1,2,...,n.
In other words,two matricesaresaid to be equal if they havethe same orderand their corresponding
entries are equal.
9
10 CHAPTER 1. MATRICES
Example 1.1.4 The linear system of equations 2x +3y = 5 and 3x+2y = 5 can be identi?ed with the
matrix
"
2 3 : 5
3 2 : 5
#
.
1.1.1 Special Matrices
De?nition 1.1.5 1. A matrix in which each entry is zero is called a zero-matrix, denoted by 0. For
example,
0
2×2
=
"
0 0
0 0
#
and 0
2×3
=
"
0 0 0
0 0 0
#
.
2. A matrix having the number of rows equal to the number of columns is called a square matrix. Thus,
its order is m×m (for some m) and is represented by m only.
3. In a square matrix, A = [a
ij
], of order n, the entries a
11
,a
22
,...,a
nn
are called the diagonal entries
and form the principal diagonal ofA.
4. A square matrix A = [a
ij
] is said to be a diagonal matrix if a
ij
= 0 for i 6= j. In other words, the
non-zero entries appear only on the principal diagonal. For example, the zero matrix 0
n
and
"
4 0
0 1
#
are a few diagonal matrices.
AdiagonalmatrixDofordernwiththediagonalentriesd
1
,d
2
,...,d
n
isdenotedbyD = diag(d
1
,...,d
n
).
If d
i
=d for all i =1,2,...,n then the diagonal matrix D is called a scalar matrix.
5. A square matrix A = [a
ij
] with a
ij
=
(
1 if i=j
0 if i6=j
is called the identity matrix, denoted byI
n
.
For example,I
2
=
"
1 0
0 1
#
, and I
3
=
?
?
?
1 0 0
0 1 0
0 0 1
?
?
?.
The subscript n is suppressed in case the order is clear from the context or if no confusion arises.
6. A square matrix A = [a
ij
] is said to be an upper triangular matrix if a
ij
=0 fori>j.
A square matrix A = [a
ij
] is said to be an lower triangular matrix if a
ij
= 0 for i<j.
A square matrix A is said to be triangular if it is an upper or a lower triangular matrix.
For example
?
?
?
2 1 4
0 3 -1
0 0 -2
?
?
? is an upper triangular matrix. An upper triangular matrix will be represented
by
?
?
?
?
?
?
a
11
a
12
··· a
1n
0 a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
0 0 ··· a
nn
?
?
?
?
?
?
.
1.2 Operations on Matrices
De?nition 1.2.1 (Transpose of a Matrix) The transpose of an m×n matrix A = [a
ij
] is de?ned as the
n×m matrix B =[b
ij
], with b
ij
=a
ji
for 1=i=m and 1=j=n. The transpose of A is denoted byA
t
.
1.2. OPERATIONS ON MATRICES 11
That is, by the transpose of an m×n matrix A, we mean a matrix of order n×m having the rows
of A as its columns and the columns of A as its rows.
For example, if A =
"
1 4 5
0 1 2
#
then A
t
=
?
?
?
1 0
4 1
5 2
?
?
?.
Thus, the transpose of a row vector is a column vector and vice-versa.
Theorem 1.2.2 For any matrix A, we have (A
t
)
t
=A.
Proof. Let A = [a
ij
], A
t
=[b
ij
] and (A
t
)
t
=[c
ij
]. Then, the de?nition of transpose gives
c
ij
=b
ji
=a
ij
for all i,j
and the result follows. 
De?nition 1.2.3 (Addition of Matrices) letA = [a
ij
] andB = [b
ij
] be are twom×n matrices. Then the
sum A+B is de?ned to be the matrix C =[c
ij
] with c
ij
=a
ij
+b
ij
.
Note that, we de?ne the sum of two matrices only when the order of the two matrices are same.
De?nition 1.2.4 (Multiplying a Scalar to a Matrix) Let A = [a
ij
] be an m×n matrix. Then for any
element k?R, we de?ne kA = [ka
ij
].
For example, if A =
"
1 4 5
0 1 2
#
and k =5, then 5A =
"
5 20 25
0 5 10
#
.
Theorem 1.2.5 Let A,B and C be matrices of order m×n, and let k,l?R. Then
1. A+B =B +A (commutativity).
2. (A+B)+C =A+(B +C) (associativity).
3. k(lA) = (kl)A.
4. (k+l)A =kA+lA.
Proof. Part 1.
Let A = [a
ij
] andB = [b
ij
]. Then
A+B = [a
ij
]+[b
ij
] = [a
ij
+b
ij
] =[b
ij
+a
ij
] = [b
ij
]+[a
ij
] =B +A
as real numbers commute.
The reader is required to prove the other parts as all the results follow from the properties of real
numbers. 
Exercise 1.2.6 1. Suppose A+B =A. Then show that B =0.
2. Suppose A+B =0. Then show that B = (-1)A = [-a
ij
].
De?nition 1.2.7 (Additive Inverse) Let A be an m×n matrix.
1. Then there exists a matrix B with A+B =0. This matrix B is called the additive inverse of A, and
is denoted by-A = (-1)A.
2. Also, for the matrix0
m×n
,A+0 =0+A=A. Hence, the matrix0
m×n
is called the additive identity.
12 CHAPTER 1. MATRICES
1.2.1 Multiplication of Matrices
De?nition 1.2.8 (Matrix Multiplication / Product) Let A = [a
ij
] be an m×n matrix and B = [b
ij
] be
an n×r matrix. The product AB is a matrix C = [c
ij
] of order m×r, with
c
ij
=
n
X
k=1
a
ik
b
kj
=a
i1
b
1j
+a
i2
b
2j
+···+a
in
b
nj
.
Observe that the productAB is de?ned if and only if
the number of columns of A = the number of rows of B.
For example, if A =
"
1 2 3
2 4 1
#
andB =
?
?
?
1 2 1
0 0 3
1 0 4
?
?
? then
AB =
"
1+0+3 2+0+0 1+6+12
2+0+1 4+0+0 2+12+4
#
=
"
4 2 19
3 4 18
#
.
Note that in this example, while AB is de?ned, the product BA is not de?ned. However, for square
matrices A andB of the same order, both the productAB and BA are de?ned.
De?nition 1.2.9 Two square matrices A and B are said to commute if AB =BA.
Remark 1.2.10 1. Note that ifA is a squarematrix of ordernthenAI
n
=I
n
A. Also for anyd?R,
the matrix dI
n
commutes with every square matrix of order n. The matrices dI
n
for any d ? R
are called scalar matrices.
2. In general, the matrix product is not commutative. For example, consider the following two
matricesA =
"
1 1
0 0
#
andB =
"
1 0
1 0
#
. Then check that the matrix product
AB =
"
2 0
0 0
#
6=
"
1 1
1 1
#
=BA.
Theorem 1.2.11 Suppose that the matrices A, B and C are so chosen that the matrix multiplications are
de?ned.
1. Then (AB)C =A(BC). That is, the matrix multiplication is associative.
2. For any k?R, (kA)B =k(AB) =A(kB).
3. ThenA(B +C) =AB +AC. That is, multiplication distributes over addition.
4. If A is an n×n matrix then AI
n
=I
n
A =A.
5. For any square matrix A of order n and D = diag(d
1
,d
2
,...,d
n
), we have
• the ?rst row of DA is d
1
times the ?rst row ofA;
• for 1=i=n, the i
th
row ofDA is d
i
times the i
th
row ofA.
A similar statement holds for the columns ofA when A is multiplied on the right byD.
Proof. Part 1. Let A = [a
ij
]
m×n
, B =[b
ij
]
n×p
and C = [c
ij
]
p×q
. Then
(BC)
kj
=
p
X
l=1
b
kl
c
lj
and (AB)
il
=
n
X
k=1
a
ik
b
kl
.
Page 5


Matrices
1.1 De?nition of a Matrix
De?nition 1.1.1 (Matrix) A rectangular array of numbers is called a matrix.
We shall mostly be concerned with matrices having real numbers as entries.
The horizontal arraysof a matrix are called itsrows and the vertical arraysare called itscolumns.
A matrix having m rows and n columns is said to have the orderm×n.
A matrix A of order m×n can be represented in the following form:
A =
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
,
where a
ij
is the entry at the intersection of the i
th
row andj
th
column.
In a more concise manner, we also denote the matrix A by [a
ij
] by suppressing its order.
Remark 1.1.2 Some books also use
?
?
?
?
?
?
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
m1
a
m2
··· a
mn
?
?
?
?
?
?
to represent a matrix.
LetA =
"
1 3 7
4 5 6
#
. Then a
11
= 1, a
12
=3, a
13
= 7, a
21
= 4, a
22
= 5, and a
23
=6.
A matrix having only one column is called a column vector; and a matrix with only one row is
called a row vector.
Whenever a vector is used, it should be understood from the context whether it is
a row vector or a column vector.
De?nition 1.1.3 (Equality of two Matrices) Two matricesA = [a
ij
] andB =[b
ij
] having the same order
m×n are equal if a
ij
=b
ij
for each i =1,2,...,m and j =1,2,...,n.
In other words,two matricesaresaid to be equal if they havethe same orderand their corresponding
entries are equal.
9
10 CHAPTER 1. MATRICES
Example 1.1.4 The linear system of equations 2x +3y = 5 and 3x+2y = 5 can be identi?ed with the
matrix
"
2 3 : 5
3 2 : 5
#
.
1.1.1 Special Matrices
De?nition 1.1.5 1. A matrix in which each entry is zero is called a zero-matrix, denoted by 0. For
example,
0
2×2
=
"
0 0
0 0
#
and 0
2×3
=
"
0 0 0
0 0 0
#
.
2. A matrix having the number of rows equal to the number of columns is called a square matrix. Thus,
its order is m×m (for some m) and is represented by m only.
3. In a square matrix, A = [a
ij
], of order n, the entries a
11
,a
22
,...,a
nn
are called the diagonal entries
and form the principal diagonal ofA.
4. A square matrix A = [a
ij
] is said to be a diagonal matrix if a
ij
= 0 for i 6= j. In other words, the
non-zero entries appear only on the principal diagonal. For example, the zero matrix 0
n
and
"
4 0
0 1
#
are a few diagonal matrices.
AdiagonalmatrixDofordernwiththediagonalentriesd
1
,d
2
,...,d
n
isdenotedbyD = diag(d
1
,...,d
n
).
If d
i
=d for all i =1,2,...,n then the diagonal matrix D is called a scalar matrix.
5. A square matrix A = [a
ij
] with a
ij
=
(
1 if i=j
0 if i6=j
is called the identity matrix, denoted byI
n
.
For example,I
2
=
"
1 0
0 1
#
, and I
3
=
?
?
?
1 0 0
0 1 0
0 0 1
?
?
?.
The subscript n is suppressed in case the order is clear from the context or if no confusion arises.
6. A square matrix A = [a
ij
] is said to be an upper triangular matrix if a
ij
=0 fori>j.
A square matrix A = [a
ij
] is said to be an lower triangular matrix if a
ij
= 0 for i<j.
A square matrix A is said to be triangular if it is an upper or a lower triangular matrix.
For example
?
?
?
2 1 4
0 3 -1
0 0 -2
?
?
? is an upper triangular matrix. An upper triangular matrix will be represented
by
?
?
?
?
?
?
a
11
a
12
··· a
1n
0 a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
0 0 ··· a
nn
?
?
?
?
?
?
.
1.2 Operations on Matrices
De?nition 1.2.1 (Transpose of a Matrix) The transpose of an m×n matrix A = [a
ij
] is de?ned as the
n×m matrix B =[b
ij
], with b
ij
=a
ji
for 1=i=m and 1=j=n. The transpose of A is denoted byA
t
.
1.2. OPERATIONS ON MATRICES 11
That is, by the transpose of an m×n matrix A, we mean a matrix of order n×m having the rows
of A as its columns and the columns of A as its rows.
For example, if A =
"
1 4 5
0 1 2
#
then A
t
=
?
?
?
1 0
4 1
5 2
?
?
?.
Thus, the transpose of a row vector is a column vector and vice-versa.
Theorem 1.2.2 For any matrix A, we have (A
t
)
t
=A.
Proof. Let A = [a
ij
], A
t
=[b
ij
] and (A
t
)
t
=[c
ij
]. Then, the de?nition of transpose gives
c
ij
=b
ji
=a
ij
for all i,j
and the result follows. 
De?nition 1.2.3 (Addition of Matrices) letA = [a
ij
] andB = [b
ij
] be are twom×n matrices. Then the
sum A+B is de?ned to be the matrix C =[c
ij
] with c
ij
=a
ij
+b
ij
.
Note that, we de?ne the sum of two matrices only when the order of the two matrices are same.
De?nition 1.2.4 (Multiplying a Scalar to a Matrix) Let A = [a
ij
] be an m×n matrix. Then for any
element k?R, we de?ne kA = [ka
ij
].
For example, if A =
"
1 4 5
0 1 2
#
and k =5, then 5A =
"
5 20 25
0 5 10
#
.
Theorem 1.2.5 Let A,B and C be matrices of order m×n, and let k,l?R. Then
1. A+B =B +A (commutativity).
2. (A+B)+C =A+(B +C) (associativity).
3. k(lA) = (kl)A.
4. (k+l)A =kA+lA.
Proof. Part 1.
Let A = [a
ij
] andB = [b
ij
]. Then
A+B = [a
ij
]+[b
ij
] = [a
ij
+b
ij
] =[b
ij
+a
ij
] = [b
ij
]+[a
ij
] =B +A
as real numbers commute.
The reader is required to prove the other parts as all the results follow from the properties of real
numbers. 
Exercise 1.2.6 1. Suppose A+B =A. Then show that B =0.
2. Suppose A+B =0. Then show that B = (-1)A = [-a
ij
].
De?nition 1.2.7 (Additive Inverse) Let A be an m×n matrix.
1. Then there exists a matrix B with A+B =0. This matrix B is called the additive inverse of A, and
is denoted by-A = (-1)A.
2. Also, for the matrix0
m×n
,A+0 =0+A=A. Hence, the matrix0
m×n
is called the additive identity.
12 CHAPTER 1. MATRICES
1.2.1 Multiplication of Matrices
De?nition 1.2.8 (Matrix Multiplication / Product) Let A = [a
ij
] be an m×n matrix and B = [b
ij
] be
an n×r matrix. The product AB is a matrix C = [c
ij
] of order m×r, with
c
ij
=
n
X
k=1
a
ik
b
kj
=a
i1
b
1j
+a
i2
b
2j
+···+a
in
b
nj
.
Observe that the productAB is de?ned if and only if
the number of columns of A = the number of rows of B.
For example, if A =
"
1 2 3
2 4 1
#
andB =
?
?
?
1 2 1
0 0 3
1 0 4
?
?
? then
AB =
"
1+0+3 2+0+0 1+6+12
2+0+1 4+0+0 2+12+4
#
=
"
4 2 19
3 4 18
#
.
Note that in this example, while AB is de?ned, the product BA is not de?ned. However, for square
matrices A andB of the same order, both the productAB and BA are de?ned.
De?nition 1.2.9 Two square matrices A and B are said to commute if AB =BA.
Remark 1.2.10 1. Note that ifA is a squarematrix of ordernthenAI
n
=I
n
A. Also for anyd?R,
the matrix dI
n
commutes with every square matrix of order n. The matrices dI
n
for any d ? R
are called scalar matrices.
2. In general, the matrix product is not commutative. For example, consider the following two
matricesA =
"
1 1
0 0
#
andB =
"
1 0
1 0
#
. Then check that the matrix product
AB =
"
2 0
0 0
#
6=
"
1 1
1 1
#
=BA.
Theorem 1.2.11 Suppose that the matrices A, B and C are so chosen that the matrix multiplications are
de?ned.
1. Then (AB)C =A(BC). That is, the matrix multiplication is associative.
2. For any k?R, (kA)B =k(AB) =A(kB).
3. ThenA(B +C) =AB +AC. That is, multiplication distributes over addition.
4. If A is an n×n matrix then AI
n
=I
n
A =A.
5. For any square matrix A of order n and D = diag(d
1
,d
2
,...,d
n
), we have
• the ?rst row of DA is d
1
times the ?rst row ofA;
• for 1=i=n, the i
th
row ofDA is d
i
times the i
th
row ofA.
A similar statement holds for the columns ofA when A is multiplied on the right byD.
Proof. Part 1. Let A = [a
ij
]
m×n
, B =[b
ij
]
n×p
and C = [c
ij
]
p×q
. Then
(BC)
kj
=
p
X
l=1
b
kl
c
lj
and (AB)
il
=
n
X
k=1
a
ik
b
kl
.
1.3. SOME MORE SPECIAL MATRICES 13
Therefore,

A(BC)


ij
=
n
X
k=1
a
ik

BC


kj
=
n
X
k=1
a
ik

p
X
l=1
b
kl
c
lj


=
n
X
k=1
p
X
l=1
a
ik

b
kl
c
lj


=
n
X
k=1
p
X
l=1

a
ik
b
kl


c
lj
=
p
X
l=1

n
X
k=1
a
ik
b
kl


c
lj
=
t
X
l=1

AB


il
c
lj
=

(AB)C


ij
.
Part 5. For allj = 1,2,...,n, we have
(DA)
ij
=
n
X
k=1
d
ik
a
kj
=d
i
a
ij
as d
ik
=0 whenever i6=k. Hence, the required result follows.
The reader is required to prove the other parts. 
Exercise 1.2.12 1. Let A and B be two matrices. If the matrix addition A+B is de?ned, then prove
that (A+B)
t
=A
t
+B
t
. Also, if the matrix product AB is de?ned then prove that (AB)
t
=B
t
A
t
.
2. Let A = [a
1
,a
2
,...,a
n
] and B =
?
?
?
?
?
?
b
1
b
2
.
.
.
b
n
?
?
?
?
?
?
. Compute the matrix products AB and BA.
3. Let n be a positive integer. Compute A
n
for the following matrices:
"
1 1
0 1
#
,
?
?
?
1 1 1
0 1 1
0 0 1
?
?
?,
?
?
?
1 1 1
1 1 1
1 1 1
?
?
?.
Can you guess a formula forA
n
and prove it by induction?
4. Find examples for the following statements.
(a) Suppose that the matrix product AB is de?ned. Then the product BA need not be de?ned.
(b) Suppose that the matrix products AB and BA are de?ned. Then the matrices AB and BA can
have di?erent orders.
(c) Suppose that the matrices A and B are square matrices of order n. Then AB and BA may or
may not be equal.
1.3 Some More Special Matrices
De?nition 1.3.1 1. A matrix A overR is called symmetric if A
t
=A and skew-symmetric if A
t
=-A.
2. A matrix A is said to be orthogonal if AA
t
=A
t
A =I.
Example 1.3.2 1. LetA =
?
?
?
1 2 3
2 4 -1
3 -1 4
?
?
? andB =
?
?
?
0 1 2
-1 0 -3
-2 3 0
?
?
?. ThenA is a symmetric matrix and
B is a skew-symmetric matrix.