Chapter 9 222 Matrices and Determinants

Chapter 9
Matrices and Determinants
9.1 Introduction:
In many economic analysis, variables are assumed to be related by
sets of linear equations. Matrix algebra provides a clear and concise
notation for the formulation and solution of such problems, many of which
would be complicated in conventional algebraic notation. The concept of
determinant and is based on that of matrix. Hence we shall first explain a
matrix.
9.2 Matrix:
A set of mn numbers (real or complex), arranged in a rectangular
formation (array or table) having m rows and n columns and enclosed by a
square bracket [ ] is called mn matrix (read “m by n matrix”) .
An mn matrix is expressed as

 a11 a12     a1n 

a a22     a2n 
 21
A=               
 
             
 am1 am2     amn 
The letters aij stand for real numbers. Note that aij is the element in
the ith row and jth column of the matrix .Thus the matrix A is sometimes
denoted by simplified form as (aij) or by {aij} i.e., A = (aij)
Matrices are usually denoted by capital letters A, B, C etc and its
elements by small letters a, b, c etc.
Order of a Matrix:
The order or dimension of a matrix is the ordered pair having as
first component the number of rows and as second component the number
of columns in the matrix. If there are 3 rows and 2 columns in a matrix,
then its order is written as (3, 2) or (3 x 2) read as three by two. In general
if m are rows and n are columns of a matrix, then its order is (m x n).
Examples:
Chapter 9 223 Matrices and Determinants

 a1 a 2 a3 a4 
1  
1 2 3  2  and  b1 b 2 b3 b 4 
4 5 6 ,    c1 c 2 c4 
   3 
c3
 
d1 d 2 d3 d4 
are matrices of orders (2 x 3), (3 x 1) and (4 x 4) respectively.
9.3 Some types of matrices:
1. Row Matrix and Column Matrix:
A matrix consisting of a single row is called a row matrix or a
row vector, whereas a matrix having single column is called a column
matrix or a column vector.

2. Null or Zero Matrix:

A matrix in which each element is „0‟ is called a Null or Zero
matrix. Zero matrices are generally denoted by the symbol O. This
distinguishes zero matrix from the real number 0.
0 0 0 0 
For example O =   is a zero matrix of order 2 x 4.
0 0 0 0 
The matrix Omxn has the property that for every matrix Amxn,
A+O=O+A=A

3. Square matrix:
A matrix A having same numbers of rows and columns is called a
square matrix. A matrix A of order m x n can be written as Amxn. If
m = n, then the matrix is said to be a square matrix. A square
matrix of order n x n, is simply written as An.

Thus and are square matrix of

order 2 and 3
Main or Principal (leading)Diagonal:
The principal diagonal of a square matrix is the ordered set of
elements aij, where i = j, extending from the upper left-hand corner to the
lower right-hand corner of the matrix. Thus, the principal diagonal
contains elements a11, a22, a33 etc.
For example, the principal diagonal of
Chapter 9 224 Matrices and Determinants

1 3 1
5 2 3 
 
6 4 0 
consists of elements 1, 2 and 0, in that order.
Particular cases of a square matrix:
(a)Diagonal matrix:
A square matrix in which all elements are zero except those in the
main or principal diagonal is called a diagonal matrix. Some elements of
the principal diagonal may be zero but not all.
1 0 0 
4 0  
For example  0 2 and 0 1 0 
  0 0 0 
are diagonal matrices.
 a11 a12    a1n 
 a a 22    a 2n 
 21
In general A =              = (a ij )nxn
 
           
 a n1 a n2    a nn 
is a diagonal matrix if and only if
aij = 0 for i  j
aij  0 for at least one i = j
(b) Scalar Matrix:
A diagonal matrix in which all the diagonal elements are same, is
called a scalar matrix i.e.
Thus
k 0 0 
and 0 k 0 are scalar matrices
 
 0 0 k 
(c) Identity Matrix or Unit matrix:
A scalar matrix in which each diagonal element is 1(unity) is
called a unit matrix. An identity matrix of order n is denoted by In.
Chapter 9 225 Matrices and Determinants

1 0 0 
1 0 
Thus I2 =   and I3 =  0 1 0 
0 1   
0 0 1 
are the identity matrices of order 2 and 3 .

 a11 a12     a1n 

a a22     a2n 
 21
In general, A=                = [aij]mxn
 
             
 am1 am2     amn 
is an identity matrix if and only if
aij = 0 for i ≠ j and aij = 1 for i = j
Note: If a matrix A and identity matrix I are comformable for
multiplication, then I has the property that
AI = IA = A i.e., I is the identity matrix for multiplication.

4. Equal Matrices:
Two matrices A and B are said to be equal if and only if they have
the same order and each element of matrix A is equal to the corresponding
element of matrix B i.e for each i, j, aij = bij

Thus A = and B =

then A = B because the order of matrices A and B is same

and aij = bij for every i , j.

Example 1: Find the values of x , y , z and a which satisfy the

matrix equation
=
Solution : By the definition of equality of matrices, we have

x + 3 = 0 ……………………………..(1)
2y + x = -7 ……………………………(2)
z – 1 = 3 ……………………………(3)
4a – 6 = 2a ……………………………(4)
From (1) x = -3
Chapter 9 226 Matrices and Determinants

Put the value of x in (2) , we get y = -2

From (3) z=4
From (4) a=3

5. The Negative of a Matrix:

The negative of the matrix Amxn, denoted by –Amxn, is the matrix
formed by replacing each element in the matrix Amxn with its additive
inverse. For example,
 3 -1
If A3x2   2 -2
-4 5 
 3 1 
Then A3x2   2 2 
 4 5
for every matrix Amxn, the matrix –Amxn has the property that
A + (–A) = (–A) + A = 0
i.e., (–A) is the additive inverse of A.
The sum Bm-n + (–Amxn) is called the difference of Bmxn and Amxn
and is denoted by Bmxn – Amxn.

9.4 Operations on matrices:

(a) Multiplication of a Matrix by a Scalar:

If A is a matrix and k is a scalar (constant), then kA is a matrix
whose elements are the elements of A , each multiplied by k
 4 3
For example, if A =  8 2 then for a scalar k,
 
 1 0 

kA =
Chapter 9 227 Matrices and Determinants

5 8 4  15 24 12 
Also, 3 0 3 5 =  0 9 15
  
3 1 4   9 3 12 

(b) Addition and subtraction of Matrices:

If A and B are two matrices of same order mn then their sum
A + B is defined as C, mn matrix such that each element of C is the sum
of the corresponding elements of A and B .
for example
If and B=

Then C=A+B = =

Similarly, the difference A – B of the two matrices A and B is a matrix

each element of which is obtained by subtracting the elements of B from
the corresponding elements of A
Thus if A= , B =

then A - B =  =

If A, B and C are the matrices of the same order mxn

then A+B = B+A
and (A + B) + C = A + (B + C) i.e., the addition of matrices is
commutative and Associative respectively.
Note: The sum or difference of two matrices of different order is not
defined.

(c) Product of Matrices:

Two matrices A and B are said to be conformable for the product
AB if the number of columns of A is equal to the number of rows of B.
Then the product matrix AB has the same number of rows as A and the
same number of columns as B.
Thus the product of the matrices Amxp and Bpxn is the matrix
(AB)mxn. The elements of AB are determined as follows:
Chapter 9 228 Matrices and Determinants

The element Cij in the ith row and jth column of (AB)mxn is found
by cij = ai1b1j  ai2b2j + ai3b3j + ……….+ ainbnj

for example, consider the matrices

 a11 a12   b11 b12 
A2x2 =   and B2x2 =  
a 21 a 22   b 21 b22 
Since the number of columns of A is equal to the number of rows of B
,the product AB is defined and is given as
 a11b11  a12 b21 a11b12  a12 b22 
AB =  
a 21b11  a 22 b21 a 21b12  a 22 b22 
Thus c11 is obtained by multiplying the elements of the first row
of A i.e., a11 , a12 by the corresponding elements of the first column of B
i.e., b11 , b21 and adding the product.
Similarly , c12 is obtained by multiplying the elements of the first
row of A i.e., a11 , a12 by the corresponding elements of the second
column of B i.e., b12 , b22 and adding the product. Similarly for c21 , c22 .
Note :
1 . Multiplication of matrices is not commutative i.e., AB  BA in
general.
2 . For matrices A and B if AB = BA then A and B commute to
each other
3 . A matrix A can be multiplied by itself if and only if it is a square
matrix.The product A.A in such cases is written as A2.
Similarly we may define higher powers of a square matrix i.e.,
A . A2 = A3 , A2. A2 = A4
4. In the product AB, A is said to be pre multiple of B and B is said
to be post multiple of A.

1 2  2 1
Example 1: If A =   and B = 1 1 Find AB and BA.
 1 3   
Solution:
1 2   2 1  22 1 2 
AB =   =  
 1 3  1 1  2  3 1  3
4 3
= 
1 2 
Chapter 9 229 Matrices and Determinants

 2 1  1 2  2  1 4  3
BA =    1 3  = 1  1
 1 1   2  3
1 7 
=  
0 5 
This example shows very clearly that multiplication of matrices in
general, is not commutative i.e., AB  BA.
Example 2: If
1 1
3 1 2   2 1  , find AB
Example 2: If A =   and B =
1 0 1   
3 1 
Solution:
Since A is a (2 x 3) matrix and B is a (3 x 2) matrix, they are
conformable for multiplication. We have
1 1
3 1 2  
AB =   = 3  2  6 3  1  2 
 2 1 1  0  3 1  0  1
1 0 1     
3 1 
11 0 
=  
 4 0
Remark:
If A, B and C are the matrices of order (m x p), (p x q) and (q x n)
respectively, then
i. (AB)C = A(BC) i.e., Associative law holds.
C(A+B) = CA + CB
ii. }i.e distributive laws holds.
and (A + B)C = AC + BC
Note: that if a matrix A and identity matrix I are conformable for
multiplication, then I has the property that
AI = IA = A i.e, I is the identity matrix for multiplication.

Exercise 9.1
Q.No. 1 Write the following matrices in tabular form:
i. A = [aij], where i = 1, 2, 3 and j = 1, 2, 3, 4
ii. B = [bij], where i = 1 and j = 1, 2, 3, 4
iii. C = [cjk], where j = 1, 2, 3 and k = 1
Chapter 9 230 Matrices and Determinants

Q.No.2 Write each sum as a single matrix:

 2 1 4  6 3 0 
i. 3 -1 0  + -2 1 0 
   
ii. 1 3 5 6 + 0 -2 1 3
4 6
iii.  3  0 
   
 1  2
 2 3 4  0 0 0 
iv.  1 6 2  0 0 0
   
 1 0 3 0 0 0
6 1 4 2
v. 2  0 3  3  0 1 
 1 2   5 1
 b11  a11 b12  a12 
Q.3 Show that   is a solution of the matrix
 b 21  a 21 b 22  a 22 
 a11 a12   b11 b12 
equation X + A = B, where A =   and B =  .
a 21 a 22   b 21 b 22 
Q.4 Solve each of the following matrix equations:
 3 1  5 1
X+  
2   3 1
i.
2
 1 0   2 6  4 8
ii. X+    1 5   2 0 
 0 2     
1 0 2  2 3 1 
iii. 3X +  2 1 3   1 2 0
   
 4 1 5   0 1 5
3 1
X + 2I = 
2 
iv.
1
Chapter 9 231 Matrices and Determinants

Q.5 Write each product as a single matrix:

1 1
3 1 1  
i. 0 1 2  0 2 
  1 0 
 
1
ii. [3 - 2 2]  2 
 
 2 
2 2 1  1 2 5 
iii. 1 1 2   1 1 3 

1 0 1  1 2 4 
 1 2 5   2 2 1
iv.  1 1 3 1 1 2 

 1 2 4 1 0 1
1 4   3 2 1 0  2
If A =  ,B=  4 0 
0 2  , find A + BC.
Q.6 ,C=
2 1   
 1 2   1 0
Q.7 Show that if A =   and B =   , then
 0 1   1 2 
(a) (A + B)(A + B)  A2 + 2AB + B2
(b) (A + B)(A – B)  A2 – B2
Q.8 Show that:
 1 2 3   a   a + 2b + 3c 
(i) 2 1 0   b    2a + b 

 3 5 1  c   3a + 5b  c 
0  sin θ   cos θ
 cos θ 0  sin θ  1 0 0 
(ii)  0
1 0   0 1 0   0 1 0

 +sin θ
0 cos θ   sin θ 0 cos θ  0 0 1 
 2 2 2   2 2 2
Q.9 If A =   and B =  
 2 2    2 2 
Show that A and B commute.