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Chapter 9

222

Matrices and Determinants

Chapter 9
Matrices and Determinants
9
...
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
...
Hence we shall first explain a
matrix
...
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”)
...
Note that aij is the element in
the ith row and jth column of the matrix
...
e
...

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
...
In general
if m are rows and n are columns of a matrix, then its order is (m x n)
...


9
...


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
...


Null or Zero Matrix:
A matrix in which each element is „0‟ is called a Null or Zero
matrix
...
This
distinguishes zero matrix from the real number 0
...

0 0 0 0 
The matrix Omxn has the property that for every matrix Amxn,
A+O=O+A=A
3
...
A matrix A of order m x n can be written as Amxn
...
A square
matrix of order n x n, is simply written as An
...
Thus, the principal diagonal
contains elements a11, a22, a33 etc
...

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
...

For example

1 0 0 
4 0


 0 2 and 0 1 0 


0 0 0 

are diagonal matrices
...
e
...
An identity matrix of order n is denoted by In
...


In general,

a12
    a1n 
 a11
a
a22
    a2n 
 21
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
...
, I is the identity matrix for multiplication
...


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
...


Example 1: Find the values of x , y , z and a which satisfy the
matrix equation
=
Solution :

From (1)

By the definition of equality of matrices, we have
x + 3 = 0 ……………………………
...


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
...
e
...

The sum Bm-n + (–Amxn) is called the difference of Bmxn and Amxn
and is denoted by Bmxn – Amxn
...
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 
3 0 3 5 =  0
9 15

 
3 1 4   9 3 12 

Also,

(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
...
e
...

Note: The sum or difference of two matrices of different order is not
defined
...

Then the product matrix AB has the same number of rows as A and the
same number of columns as B
...
The elements of AB are determined as follows:

Chapter 9

by

228

Matrices and Determinants

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

Similarly , c12 is obtained by multiplying the elements of the first
row of A i
...
, a11 , a12 by the corresponding elements of the second
column of B i
...
, b12 , b22 and adding the product
...


Note :

1
...
e
...

2
...
A matrix A can be multiplied by itself if and only if it is a square
matrix
...
A in such cases is written as A2
...
e
...
A2 = A3 , A2
...
In the product AB, A is said to be pre multiple of B and B is said
to be post multiple of A
...

 1 3 



Solution:

1 2 

AB = 

= 

 1 3  1 1  2  3 1  3

4 3
2 

= 
1

Chapter 9

229

 2 1  1

Matrices and Determinants

2

 2  1 4  3
2  3

BA = 
  1 3  = 1  1
1
1


 

1 7 

= 

0 5 
This example shows very clearly that multiplication of matrices in
general, is not commutative i
...
, AB  BA
...
We have

1 1
3 1 2  
 = 3  2  6 3  1  2 
AB = 
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
...
e
...

ii
...
e distributive laws holds
...
e, I is the identity matrix for multiplication
...
1
Q
...
1 Write the following matrices in tabular form:
i
...
B = [bij], where i = 1 and j = 1, 2, 3, 4
iii
...
No
...

ii
...


iv
...


Q
...


 b 21 b 22 
a 21 a 22 

equation X + A = B, where A = 
Q
...


X+ 
2

ii
...


iv
...
5
i
...


iii
...


231

Matrices and Determinants

Write each product as a single matrix:

1 1
3 1 1 

0 1 2  0 2 

 1 0 


1
[3 - 2 2]  2 
 
 2 
2
1

1
 1
 1

 1

2 1  1
1 2   1
0 1  1
2 5   2
1 3 1
2 4 1

 3 2 
,C=
0 

Q
...



 1 2 
 1 0
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
...
6

1 4 

2 5 
1 3 
2 4 
2 1
1 2 
0 1

If A = 
,B=  4
2 1


(i)

Q
...


0  sin θ  1 0 0 
1
0   0 1 0
0 cos θ  0 0 1 
2 2

2 

Chapter 9

232

Matrices and Determinants

Answers 9
...
1(i)

(ii)
(iii)

Q
...
4

(i)

(iii)

Q
...
6

9
...
The determinants are defined only for square matrices
...

The determinant of the (2 x 2) matrix

 a11 a12 

a 21 a 22 

A= 

is given by det A = |A| =

a11 a12
a 21 a 22

= a11 a22 – a12 a21
Example 3:

3

1

If A = 
 find |A|
 2 3

Solution:
|A| =

3 1
= 9 – (–2) = 9 + 2 = 11
2 3

The determinant of the (3 x 3) matrix

 a11 a12
A = a 21 a 22

 a 31 a 32

a13 
a11 a12

a 23  , denoted by |A| = a 21 a 22
a 33 
a 31 a 32

a13
a 23
a 33

is given as, det A = |A|
= a11

a 22
a 32

a 23
a 21 a 23
a 21 a 22
– a12
+ a13
a 33
a 31 a 33
a 31 a 32

= a11(a22a33 – a23a32) – a12(a21a33 – a23a31) + a13(a21a32 – a22a31)
Note: Each determinant in the sum (In the R
...
S) is the determinant of a
submatrix of A obtained by deleting a particular row and column of A
...
We take the sign + or  , according
to (  1)i+j aij
Where i and j represent row and column
...
6
Minor and Cofactor of Element:
The minor Mij of the element aij in a given determinant is the
determinant of order (n – 1 x n – 1) obtained by deleting the ith row and
jth column of Anxn
...

a 31 a 32

The scalars Cij = (-1)i+j Mij are called the cofactor of the element aij
of the matrix A
...

The value of the determinant can be found by expanding it from
any row or column
...

Solution (a)
|A|

3 2 1
= 0 1 2
1 3 4

Chapter 9

235

=3

(b)

Matrices and Determinants

1 2
0 2
0 1
–2
+1
3 4
1 4
1 3

|A|

= 3(4 + 6) – 2(0 + 2) + 1 (0 – 1)
= 30 – 4 – 1
= 25

|A|

=3

1 2
2 1
0 1
–0
+1
3 4
3 4
1 3

= 3 (4 + 6) + 1 (–4 –1)
= 30 – 5
|A|
= 25
9
...
Interchanging the corresponding rows and columns of a
determinant does not change its value (i
...
, |A| = |A‟|)
...
(2)
Now again consider
|B|

a1 a 2
= b1 b 2
c1 c2

a3
b3
c3

Expand it by first column
|B|
= a1(b2c3 – b3c2) – b1(a2c3 – a3c2) + c1(a2b3 – a3b2)
which is same as equation (2)
so

a1
|B| = a 2
a3

b1
b2
b3

c1
c2
c3

or
|B| = |A|
2
...

For example if

Chapter 9

236

|A|

=

a1
a2
a3

Matrices and Determinants

b1
b2
b3

c1
c2
c3

b2
b1
b3

c2
c1
c3

Consider the determinant,
|B|

=

a2
a1
a3

expand by second row,
|B|
= –a1(b2c3 – b3c2) + b1(a2c3 – a3c2) – c1(a2b3 – a3b2)
= –(a1(b2c3 – b3c2) – b1(a2c3 – a3c2) + c1(a2b3 – a3b2))
The term in the bracket is same as the equation (2)
So

|B|

=

a1
– a2
a3

b1
b2
b3

c1
c2
c3

Or
|B|
= – |A|
3
...
For example
|A|

=

0
a2
a3

0
b2
b3

0
c2
c3

=
0(b2c3 – b3c2) –0(a2c3 – a3c2) +0(a2b3 – a3b3)
|A|
=
0
4
...
For example, if
|A|

=

a1
a1
a3

b1 c1
b1 c1
b 3 c3

= a1(b1c3 – b3c1) – b1(a1c3 – a3c1) + c1(a1b3 – a3b1)
= a1b1c3 – a1b3c1 – a1b1c3 + a3b1c1 + a1b3c1 – a3b1c1
|A|
=
0
5
...
For example if,

Chapter 9

237

|A|

=

Matrices and Determinants

a1
a2
a3

b1
b2
b3

c1
c2
c3

ka1 kb1 kc1
Consider a determinant, |B| = a 2
b2 c2
a 3 b 3 c3
|B| = ka1(b2c3 – b3c3) – kb1(a2c3 – a3c2) + kc1(a2b3 – a3b2)
= k(a1(b2c3 – b3c3) – b1(a2c3 – a3c2) + c1(a2b3 – a3b2))
So

|B|

a1
= k a2
a3

b1
b2
b3

c1
c2
c3

Or
|B|
=
K|A|
6
...

For example, if
|A|

=

a1
a2
a3

b1
b2
b3

c1
c2
c3

Consider a matrix,
|B|

=

a1  ka 2
a2
a3

b1  kb2
b2
b3

c1 +kc2
c2
c3

= (a1+ka2)(b2c3–b3c2)–(b1+kb2)(a2c3–a3c2)+(c1+kc2)(a2b3–a3b2)
= [a1(b2c3 – b3c2) –b1(a2c3 – a3c2) +c1(a2b3 – a3b2)]
= [ka2(b2c3 – b3c2) –kb2(a2c3 – a3c2) +kc2(a2b3 – a3b2)]

a1
= a2
a3

b1
b2
b3

c1
a2
c2 + k a 2
c3
a3

b2
b2
b3

c2
c2
c3

Chapter 9

a1
= a2
a3

238

b1
b2
b3

Matrices and Determinants

c1
c2 + k(0) because row 1st and 2nd are identical
c3

|B| = |A|
7
...
For example, if
|A|

2 0 0
= 0 5 0
0 0 3

= 2(–15 – 0) – (0 – 0) + 0(0 – 0)
= 30, which is the product of diagonal elements
...
e
...
The determinant of the product of two matrices is equal to the
product of the

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