1. The document discusses matrices and determinants. It defines different types of matrices such as rectangular, square, diagonal, scalar, row, column, identity, zero, upper triangular, and lower triangular matrices. 2. It explains how to calculate determinants of matrices. The determinant of a 1x1 matrix is the single element. The determinant of a 2x2 matrix is calculated using a formula. Determinants of higher order matrices are calculated by expanding along rows or columns. 3. It introduces concepts of minors, cofactors, and explains how the value of a determinant can be written in terms of its minors and cofactors. It also lists some properties and operations for determinants.

1. Matrices & Determinants
Determinants
Dr. Pankaj Das
Mail id: Pankaj.iasri@gmail.com

2. Matrices & Determinants
TYPES OF MATRICES
NAME DESCRIPTION EXAMPLE
Rectangular
matrix
No. of rows is not equal to
no. of columns
Square matrix No. of rows is equal to no. of
columns
Diagonal
matrix
Non-zero element in principal
diagonal and zero in all other
positions
Scalar matrix Diagonal matrix in which all
the elements on principal
diagonal and same








5
0
2
1
2
6

2 1 3
2 0 1
1 2 4




















7
0
0
0
4
0
0
0
2










4
0
0
0
4
0
0
0
4

3. Matrices & Determinants
TYPES OF MATRICES
NAME DESCRIPTION EXAMPLE
Row matrix A matrix with only 1
row
Column matrix A matrix with only I
column
Identity matrix Diagonal matrix
having each
diagonal element
equal to one (I)
Zero matrix A matrix with all zero
entries

3 2 14
 

2
3












1
0
0
1
0 0
0 0







4. Matrices & Determinants
TYPES OF MATRICES
NAME DESCRIPTION EXAMPLE
Upper Triangular
matrix
Square matrix
having all the entries
zero below the
principal diagonal
Lower Triangular
matrix
Square matrix
having all the entries
zero above the
principal diagonal










7
0
0
6
4
0
3
5
2










7
3
6
0
4
5
0
0
2

5. Matrices & Determinants
Determinants
If is a square matrix of order 1,
then |A| = | a11 | = a11
ij
A = a
 
 
If is a square matrix of order 2, then
11 12
21 22
a a
A =
a a
 
 
 
|A| = = a11a22 – a21a12
a a
a a
1
1 1
2
2
1 2
2

6. Matrices & Determinants
Example
4 - 3
Evaluate the determinant :
2 5
 
4 - 3
Solution : = 4 ×5 - 2 × -3 = 20 + 6 = 26
2 5

7. Matrices & Determinants
Solution
If A = is a square matrix of order 3, then
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
 
 
 
 
 
[Expanding along first row]
11 12 13
22 23 21 23 21 22
21 22 23 11 12 13
32 33 31 33 31 32
31 32 33
a a a
a a a a a a
| A |= a a a = a - a + a
a a a a a a
a a a
     
11 22 33 32 23 12 21 33 31 23 13 21 32 31 22
= a a a - a a - a a a - a a + a a a - a a
   
11 22 33 12 31 23 13 21 32 11 23 32 12 21 33 13 31 22
a a a a a a a a a a a a a a a a a a
     

8. Matrices & Determinants
Example
2 3 - 5
Evaluate the determinant : 7 1 - 2
-3 4 1
 
2 3 - 5
1 - 2 7 - 2 7 1
7 1 - 2 = 2 - 3 + -5
4 1 -3 1 -3 4
-3 4 1
     
= 2 1+ 8 - 3 7 - 6 - 5 28 + 3
= 18 - 3 - 155
= -140
[Expanding along first row]
Solution :

9. Matrices & Determinants
Minors
-1 4
If A = , then
2 3
 
 
 
21 21 22 22
M = Minor of a = 4, M = Minor of a = -1
11 11 12 12
M = Minor of a = 3, M = Minor of a = 2

10. Matrices & Determinants
Minors
4 7 8
If A = -9 0 0 , then
2 3 4
 
 
 
 
 
M11 = Minor of a11 = determinant of the order 2 × 2 square
sub-matrix is obtained by leaving first
row and first column of A
0 0
= =0
3 4
Similarly, M23 = Minor of a23
4 7
= =12-14=-2
2 3
M32 = Minor of a32 etc.
4 8
= =0+72=72
-9 0

11. Matrices & Determinants
Cofactors
 i+j
ij ij ij
C = Cofactor of a in A = -1 M ,
ij ij
where M is minor of a in A

12. Matrices & Determinants
Cofactors (Con.)
C11 = Cofactor of a11 = (–1)1 + 1 M11 = (–1)1 +1
0 0
=0
3 4
C23 = Cofactor of a23 = (–1)2 + 3 M23 =  
4 7
2
2 3
C32 = Cofactor of a32 = (–1)3 + 2M32 = etc.
4 8
- =-72
-9 0
4 7 8
A = -9 0 0
2 3 4
 
 
 
 
 

13. Matrices & Determinants
Value of Determinant in Terms of
Minors and Cofactors
11 12 13
21 22 23
31 32 33
a a a
If A = a a a , then
a a a
 
 
 
 
 
 
3 3
i j
ij ij ij ij
j 1 j 1
A 1 a M a C

 
  
 
i1 i1 i2 i2 i3 i3
= a C +a C +a C , for i=1 or i=2 or i=3

14. Matrices & Determinants
Properties of Determinants
1. The value of a determinant remains unchanged, if its
rows and columns are interchanged.
1 1 1 1 2 3
2 2 2 1 2 3
3 3 3 1 2 3
a b c a a a
a b c = b b b
a b c c c c
i e A A

. . '
2. If any two rows (or columns) of a determinant are interchanged,
then the value of the determinant is changed by minus sign.
 
1 1 1 2 2 2
2 2 2 1 1 1 2 1
3 3 3 3 3 3
a b c a b c
a b c = - a b c R R
a b c a b c
Applying 

15. Matrices & Determinants
Properties (Con.)
3. If all the elements of a row (or column) is multiplied by a
non-zero number k, then the value of the new determinant
is k times the value of the original determinant.
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 3
ka kb kc a b c
a b c = k a b c
a b c a b c
which also implies
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 3
a b c ma mb mc
1
a b c = a b c
m
a b c a b c

16. Matrices & Determinants
Properties (Con.)
4. If each element of any row (or column) consists of
two or more terms, then the determinant can be
expressed as the sum of two or more determinants.
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3
a +x b c a b c x b c
a +y b c = a b c + y b c
a +z b c a b c z b c
5. The value of a determinant is unchanged, if any row
(or column) is multiplied by a number and then added
to any other row (or column).
 
1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 1 1 2 3
3 3 3 3 3 3 3 3
a b c a +mb - nc b c
a b c = a +mb - nc b c C C + mC -nC
a b c a +mb - nc b c
Applying 

17. Matrices & Determinants
Properties (Con.)
6. If any two rows (or columns) of a determinant are
identical, then its value is zero.
2 2 2
3 3 3
0 0 0
a b c = 0
a b c
7. If each element of a row (or column) of a determinant is zero,
then its value is zero.
1 1 1
2 2 2
1 1 1
a b c
a b c = 0
a b c

18. Matrices & Determinants
Properties (Con.)
 
a 0 0
8 Let A = 0 b 0 be a diagonal matrix, then
0 0 c
 
 
 
 
 
a 0 0
= 0 b 0
0 0 c
A abc


19. Matrices & Determinants
Row(Column) Operations
Following are the notations to evaluate a determinant:
Similar notations can be used to denote column
operations by replacing R with C.
(i) Ri to denote ith row
(ii) Ri Rj to denote the interchange of ith and jth
rows.
(iii) Ri Ri + lRj to denote the addition of l times the
elements of jth row to the corresponding elements
of ith row.
(iv) lRi to denote the multiplication of all elements of
ith row by l.



20. Matrices & Determinants
Evaluation of Determinants
If a determinant becomes zero on putting
is the factor of the determinant.
 
x = , then x -
 
2
3
x 5 2
For example, if Δ = x 9 4 , then at x =2
x 16 8
, because C1 and C2 are identical at x = 2
Hence, (x – 2) is a factor of determinant .
  0


21. Matrices & Determinants
Sign System for Expansion of
Determinant
Sign System for order 2 and order 3 are
given by
+ – +
+ –
, – + –
– +
+ – +

22. Matrices & Determinants
 
42 1 6 6×7 1 6
i 28 7 4 = 4×7 7 4
14 3 2 2×7 3 2
 
1
6 1 6
=7 4 7 4 Taking out 7 common from C
2 3 2
Example-1
6 -3 2
2 -1 2
-10 5 2
42 1 6
28 7 4
14 3 2
Find the value of the following determinants
(i) (ii)
Solution :
1 3
= 7 × 0 C and C are identical
= 0
 
 

23. Matrices & Determinants
Example –1 (ii)
6 -3 2
2 -1 2
-10 5 2
(ii)
 
 
 
   
    
 
3 2 3 2
1 2 1 2
5 2 5 2
 
    
 
 
    
 

1
1 2
3 3 2
( 2) 1 1 2 Taking out 2 common from C
5 5 2
( 2) 0 C and C are identical
0

24. Matrices & Determinants
Evaluate the determinant
1 a b+c
1 b c+a
1 c a+b
Solution :
 
3 2 3
1 a b+c 1 a a+b+c
1 b c+a = 1 b a+b+c Applying c c +c
1 c a+b 1 c a+b+c

    3
1 a 1
= a+b+c 1 b 1 Taking a+b+c common from C
1 c 1
 
 
Example - 2
  1 3
= a + b + c × 0 C and C are identical
= 0
 
 

25. Matrices & Determinants
2 2 2
a b c
We have a b c
bc ca ab
 
2
1 1 2 2 2 3
(a-b) b-c c
= (a-b)(a+b) (b-c)(b+c) c Applying C C -C and C C -C
-c(a-b) -a(b-c) ab
 
   
2
1 2
1 1 c
Taking a-b and b-c common
=(a-b)(b-c) a+b b+c c
from C and C respectively
-c -a ab
 
 
 
Example - 3
bc
2 2 2
a b c
a b c
ca ab
Evaluate the determinant:
Solution:

26. Matrices & Determinants
 
2
1 1 2
0 1 c
=(a-b)(b-c) -(c-a) b+c c Applying c c -c
-(c-a) -a ab

2
0 1 c
=-(a-b)(b-c)(c-a) 1 b+c c
1 -a ab
 
2
2 2 3
0 1 c
= -(a-b)(b-c)(c-a) 0 a+b+c c -ab Applying R R -R
1 -a ab

Now expanding along C1 , we get
(a-b) (b-c) (c-a) [- (c2 – ab – ac – bc – c2)]
= (a-b) (b-c) (c-a) (ab + bc + ac)
Solution Cont.

27. Matrices & Determinants
Without expanding the determinant,
prove that
3
3x+y 2x x
4x+3y 3x 3x =x
5x+6y 4x 6x
3x+y 2x x 3x 2x x y 2x x
L.H.S= 4x+3y 3x 3x = 4x 3x 3x + 3y 3x 3x
5x+6y 4x 6x 5x 4x 6x 6y 4x 6x
3 2
3 2 1 1 2 1
= x 4 3 3 +x y 3 3 3
5 4 6 6 4 6
Example-4
Solution :
 
3 2
1 2
3 2 1
= x 4 3 3 +x y×0 C and C are identical in II determinant
5 4 6

28. Matrices & Determinants
Solution Cont.
 
3
1 1 2
1 2 1
= x 1 3 3 Applying C C -C
1 4 6

 
3
2 2 1 3 3 2
1 2 1
=x 0 1 2 ApplyingR R -R and R R -R
0 1 3
 
 
3
1
3
= x ×(3-2) Expanding along C
=x = R.H.S.
3
3 2 1
=x 4 3 3
5 4 6

29. Matrices & Determinants
Prove that : = 0 , where wis cube root of unity.
3 5
3 4
5 5
1 ω ω
ω 1 ω
ω ω 1
3 5 3 3 2
3 4 3 3
5 5 3 2 3 2
1 ω ω 1 ω ω .ω
L.H.S= ω 1 ω = ω 1 ω .ω
ω ω 1 ω .ω ω .ω 1
 
2
3
2 2
1 2
1 1 ω
= 1 1 ω ω =1
ω ω 1
=0=R.H.S. C and C are identical
 
 
Example -5
Solution :

30. Matrices & Determinants
Example-6
2
x+a b c
a x+b c =x (x+a+b+c)
a b x+C
Prove that :
 
1 1 2 3
x+a b c x+a+b+c b c
L.H.S= a x+b c = x+a+b+c x+b c
a b x+C x+a+b+c b x+c
Applying C C +C +C

Solution :
 
  1
1 b c
= x+a+b+c 1 x+b c
1 b x+c
Taking x+a+b+c commonfrom C
 
 

31. Matrices & Determinants
Solution cont.
 
2 2 1 3 3 1
1 b c
=(x+a+b+c) 0 x 0
0 0 x
Applying R R -R and R R -R
 
Expanding along C1 , we get
(x + a + b + c) [1(x2)] = x2 (x + a + b + c)
= R.H.S

32. Matrices & Determinants
 
1 1 2 3
2(a+b+c) 2(a+b+c) 2(a+b+c)
= c+a a+b b+c Applying R R +R +R
a+b b+c c+a

1 1 1
=2(a+b+c) c+a a+b b+c
a+b b+c c+a
Example -7
Solution :
Using properties of determinants, prove that
2 2 2
b+c c+a a+b
c+a a+b b+c =2(a+b+c)(ab+bc+ca-a -b -c ).
a+b b+c c+a
b+c c+a a+b
L.H.S= c+a a+b b+c
a+b b+c c+a

33. Matrices & Determinants
 
1 1 2 2 2 3
0 0 1
=2(a+b+c)(c-b) (a-c) b+c Applying C C -C and C C -C
(a-c) (b-a) c+a
 
Now expanding along R1 , we get
2
2(a+b+c) (c-b)(b-a)-(a-c)
 
 
2 2 2
=2(a+b+c) bc -b - ac+ab-(a +c -2ac)
 
 
Solution Cont.
2 2 2
=2(a+b+c) ab+bc+ac-a -b -c
=R.H.S
 
 

34. Matrices & Determinants
Using properties of determinants prove that
2
x+4 2x 2x
2x x+4 2x =(5x+4)(4-x)
2x 2x x+4
Example - 8
1 2x 2x
=(5x+4)1 x+4 2x
1 2x x+4
Solution :
 
1 1 2 3
x+4 2x 2x 5x+4 2x 2x
L.H.S= 2x x+4 2x =5x+4 x+4 2x Applying C C +C +C
2x 2x x+4 5x+4 2x x+4


35. Matrices & Determinants
Solution Cont.
 
2 2 1 3 3 2
1 2x 2x
=(5x+4) 0 -(x-4) 0 ApplyingR R -R and R R -R
0 x-4 -(x-4)
 
Now expanding along C1 , we get
2
(5x+4) 1(x- 4) -0
 
 
2
=(5x+4)(4-x)
=R.H.S

36. Matrices & Determinants
Example -9
Using properties of determinants, prove that
x+9 x x
x x+9 x =243 (x+3)
x x x+9
x+9 x x
L.H.S= x x+9 x
x x x+9
 
1 1 2 3
3x+9 x x
= 3x+9 x+9 x Applying C C +C +C
3x+9 x x+9

Solution :

37. Matrices & Determinants
 
1
=3(x+3) 81 Expanding along C
=243(x+3)
=R.H.S.

1 x x
=(3x+9)1 x+9 x
1 x x+9
Solution Cont.
  2 2 1 3 3 2
1 x x
=3 x+3 0 9 0 Applying R R -R and R R -R
0 -9 9
 
 
 

38. Matrices & Determinants
Example -10
Solution :
2 2 2 2 2
2 2 2 2 2
1 1 3
2 2 2 2 2
(b+c) a bc b +c a bc
L.H.S.= (c+a) b ca = c +a b ca Applying C C -2C
(a+b) c ab a +b c ab
 

 
 
2 2 2 2
2 2 2 2
1 1 2
2 2 2 2
a +b +c a bc
a +b +c b ca Applying C C +C
a +b +c c ab
 
2
2 2 2 2
2
1 a bc
=(a +b +c )1 b ca
1 c ab
2 2
2 2 2 2 2
2 2
(b+c) a bc
(c+a) b ca =(a +b +c )(a-b)(b-c)(c-a)(a+b+c)
(a+b) c ab
Show that

39. Matrices & Determinants
Solution Cont.
 
2
2 2 2
2 2 1 3 3 2
1 a bc
=(a +b +c ) 0 (b-a)(b+a) c(a-b) Applying R R -R and R R -R
0 (c-b)(c+b) a(b-c)
 
 
2 2 2 2 2
1
=(a +b +c )(a-b)(b-c)(-ab-a +bc+c ) Expanding along C
2 2 2
=(a +b +c )(a-b)(b-c)(c-a)(a+b+c)=R.H.S.
2
2 2 2
1 a bc
=(a +b +c )(a-b)(b-c) 0 -(b+a) c
0 -(b+c) a
    
2 2 2
=(a +b +c )(a-b)(b-c) b c-a + c-a c+a
 
 

40. Matrices & Determinants
Applications of Determinants
(Area of a Triangle)
The area of a triangle whose vertices are
is given by the expression
1 1 2 2 3 3
(x , y ), (x , y ) and (x , y )
1 1
2 2
3 3
x y 1
1
Δ= x y 1
2
x y 1
1 2 3 2 3 1 3 1 2
1
= [x (y - y ) + x (y - y ) + x (y - y )]
2

41. Matrices & Determinants
Example
Find the area of a triangle whose
vertices are (-1, 8), (-2, -3) and (3, 2).
Solution :
1 1
2 2
3 3
x y 1 -1 8 1
1 1
Area of triangle= x y 1 = -2 -3 1
2 2
x y 1 3 2 1
 
1
= -1(-3-2)-8(-2-3)+1(-4+9)
2
 
1
= 5+40+5 =25 sq.units
2

42. Matrices & Determinants
Condition of Collinearity of Three
Points
If are three points,
then A, B, C are collinear
1 1 2 2 3 3
A (x , y ), B (x , y ) and C (x , y )
1 1 1 1
2 2 2 2
3 3 3 3
Area of triangle ABC = 0
x y 1 x y 1
1
x y 1 = 0 x y 1 = 0
2
x y 1 x y 1

 

43. Matrices & Determinants
If the points (x, -2) , (5, 2), (8, 8) are collinear,
find x , using determinants.
Example
Solution :
x -2 1
5 2 1 =0
8 8 1

      
x 2-8 - -2 5-8 +1 40-16 =0

-6x-6+24=0

6x=18 x=3
 
Since the given points are collinear.

44. Matrices & Determinants
Solution of System of 2 Linear
Equations (Cramer’s Rule)
Let the system of linear equations be
 
2 2 2
a x+b y = c ... ii
 
1 1 1
a x+b y = c ... i
1 2
D D
Then x = , y = provided D 0,
D D

1 1 1 1 1 1
1 2
2 2 2 2 2 2
a b c b a c
where D = , D = and D =
a b c b a c

45. Matrices & Determinants
Cramer’s Rule (Con.)
then the system is consistent and has infinitely many
solutions.
  1 2
2 If D = 0 and D = D = 0,
then the system is inconsistent and has no solution.
 
1 If D 0
Note :
,

then the system is consistent and has unique solution.
  1 2
3 If D=0 and one of D , D 0,


46. Matrices & Determinants
Example
2 -3
D= =2+9=11 0
3 1

1
7 -3
D = =7+15=22
5 1
2
2 7
D = =10-21=-11
3 5
Solution :
1 2
D 0
D D
22 -11
By Cramer's Rule x= = =2 and y= = =-1
D 11 D 11


Using Cramer's rule , solve the following
system of equations 2x-3y=7, 3x+y=5

47. Matrices & Determinants
Solution of System of 3 Linear
Equations (Cramer’s Rule)
Let the system of linear equations be
 
2 2 2 2
a x+b y+c z = d ... ii
 
1 1 1 1
a x+b y+c z = d ... i
 
3 3 3 3
a x+b y+c z = d ... iii
3
1 2 D
D D
Then x = , y = z = provided D 0,
D D D
, 
1 1 1 1 1 1 1 1 1
2 2 2 1 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3
a b c d b c a d c
where D = a b c , D = d b c , D = a d c
a b c d b c a d c
1 1 1
3 2 2 2
3 3 3
a b d
and D = a b d
a b d

48. Matrices & Determinants
Cramer’s Rule (Con.)
Note:
(1) If D  0, then the system is consistent and has a unique
solution.
(2) If D=0 and D1 = D2 = D3 = 0, then the system has infinite
solutions or no solution.
(3) If D = 0 and one of D1, D2, D3  0, then the system
is inconsistent and has no solution.
(4) If d1 = d2 = d3 = 0, then the system is called the system of
homogeneous linear equations.
(i) If D  0, then the system has only trivial solution x = y = z = 0.
(ii) If D = 0, then the system has infinite solutions.

49. Matrices & Determinants
Example
Using Cramer's rule , solve the following
system of equations
5x - y+ 4z = 5
2x + 3y+ 5z = 2
5x - 2y + 6z = -1
Solution :
5 -1 4
D= 2 3 5
5 -2 6
1
5 -1 4
D = 2 3 5
-1 -2 6
= 5(18+10)+1(12+5)+4(-4 +3)
= 140 +17 –4
= 153
= 5(18+10) + 1(12-25)+4(-4 -15)
= 140 –13 –76 =140 - 89
= 51 0


50. Matrices & Determinants
3
5 -1 5
D = 2 3 2
5 -2 -1
= 5(-3 +4)+1(-2 - 10)+5(-4-15)
= 5 – 12 – 95 = 5 - 107
= - 102
Solution Cont.
1 2
3
D 0
D D
153 102
By Cramer's Rule x = = =3, y = = =2
D 51 D 51
D -102
and z= = =-2
D 51


2
5 5 4
D = 2 2 5
5 -1 6
= 5(12 +5)+5(12 - 25)+ 4(-2 - 10)
= 85 + 65 – 48 = 150 - 48
= 102

51. Matrices & Determinants
Example
Solve the following system of homogeneous linear equations:
x + y – z = 0, x – 2y + z = 0, 3x + 6y + -5z = 0
Solution:
     
1 1 - 1
We have D = 1 - 2 1 = 1 10 - 6 - 1 -5 - 3 - 1 6 + 6
3 6 - 5
= 4 + 8 - 12 = 0
 
 
 
 
 
The systemhas infinitely many solutions.

Putting z = k, in first two equations, we get
x + y = k, x – 2y = -k

52. Matrices & Determinants
Solution (Con.)
1
k 1
D -k - 2 -2k + k k
By Cramer's rule x = = = =
D -2 - 1 3
1 1
1 - 2

2
1 k
D 1 - k -k - k 2k
y = = = =
D -2 - 1 3
1 1
1 - 2
k 2k
x = , y = , z = k , where k R
3 3
 
These values of x, y and z = k satisfy (iii) equation.

53. Matrices & Determinants
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