Determinants
Matrices & Determinants
Objectives
•
•
•
Minor & co-factors
Computation of adjoint and inverse of a
matrix
Solving a system of linear equations
Matrices & Determinants
Determinants
If A = aij  is a square matrix of order 1,
then |A| = | a11 | = a11
 a11 a12 
A
=
If
a

 21 a22 
|A| =
a11
a12
a21
a22
is a square matrix of order 2, then
= a11a22 – a21a12
Matrices & Determinants
Example
Evaluate the determinant :
4 -3
2
5
4 -3
Solution :
= 4 × 5 - 2 × -3  = 20 + 6 = 26
2
5
Matrices & Determinants
Matrices & Determinants
Solution
a11

If A = a21
a31
a11
a12
a22
a32
a12
| A |= a21 a22
a31 a32
a13 
a23  is a square matrix of order 3, then
a33 
a13
a
a23 = a11 22
a32
a33
a23
a
a23
a
a22
- a12 21
+ a13 21
a33
a31 a33
a31 a32
[Expanding along first row]
= a11 a22a33 - a32a23  - a12 a21a33 - a31a23  + a13 a21a32 - a31a22 
  a11a22 a33  a12 a31a23  a13 a21a32   a11a23 a32  a12 a21a33  a13 a31a22 
Matrices & Determinants
Example
2
3
Evaluate the determinant : 7
1
-2
4
1
-3
-5
Solution :
2
3
7
-3
1
4
-5
1
-2 =2
4
1
-2
7
-3
1
-3
-2
7
+ -5
1
-3
1
4
[Expanding along first row]
= 2 1 + 8  - 3 7 - 6  - 5 28 + 3 
= 18 - 3 - 155
= -140
Matrices & Determinants
Minors
-1
If A =
2
4
, then

3
M11 = Minor of a11 = 3, M12 = Minor of a12 = 2
M21 = Minor of a21 = 4, M22 = Minor of a22 = -1
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
3 4
=0
Similarly, M23 = Minor of a23 =
M32 = Minor of a32 =
4
8
-9 0
4 7
2 3
=12 -14 = -2
= 0+72 = 72 etc.
Matrices & Determinants
Cofactors
Cij = Cofactor of aij in A = -1
i+ j
Mij ,
where Mij is minor of aij in A
Matrices & Determinants
Cofactors (Con.)
 4 7 8
A = -9 0 0 
 2 3 4 
C11 = Cofactor of a11 = (–1)1 + 1 M11 = (–1)1 +1
C23 = Cofactor of a23 =
(–1)2 + 3
0 0
3 4
=0
4 7
M23 = 
2
2 3
C32 = Cofactor of a32 = (–1)3 + 2M32 = -
4
8
-9 0
= - 72 etc.
Matrices & Determinants
Value of Determinant in Terms
of Minors and Cofactors
a11
If A = a21
a31
3
A 
a12
a22
a32
  1
j1
i j
a13 
a23  , then
a33 
3
aijMij 
 aijCij
j1
= ai1Ci1 + ai2Ci2 + ai3Ci3 , for i =1 or i = 2 or i = 3
Matrices & Determinants
Properties of Determinants
1.
The value of a determinant remains unchanged, if its
rows and columns are interchanged.
a1
b1
a2 b2
a3 b3
2.
c1
a1
a2
a3
c2 = b1 b2 b3
c3
c1 c2 c3
i.e. A  A '
If any two rows (or columns) of a determinant are interchanged,
then the value of the determinant is changed by minus sign.
a1
b1
a2 b2
a3 b3
c1
a2 b2
c2
c2 = - a1 b1
c3
a3 b3
c1
c3
 Applying
R2  R1 
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.
ka1 kb1 kc1
a1
b1
c1
a2
b2
c2
= k a2 b2
c2
a3
b3
c3
a3 b3
c3
which also implies
a1 b1
c1
ma1 mb1 mc1
1
a2 b2 c2 =
a2
m
a3 b3 c3
a3
b2
b3
c2
c3
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.
a1 + x b1
c1
a2 + y b2
c2 = a2 b2
c3
a3 b3
a3 + z b3
5.
a1
b1
c1
x b1
c1
c2 + y b2
c2
c3
c3
z b3
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).
a1
b1
a2 b2
a3 b3
c1
a1 + mb1 - nc1
b1
c1
c2 = a2 + mb2 - nc2 b2
c3
a3 + mb3 - nc3 b3
c2
 Applying C1  C1 + mC2 - nC3 
c3
Matrices & Determinants
Properties (Con.)
6.
If any two rows (or columns) of a determinant are
identical, then its value is zero.
a1
b1
c1
a2 b2
c2 = 0
a1
c1
b1
7. If each element of a row (or column) of a determinant is zero,
then its value is zero.
0
0
0
a2 b2
c2 = 0
a3 b3
c3
Matrices & Determinants
Properties (Con.)
8
a 0

Let A = 0 b
0 0
a 0
0

0 be a diagonal matrix, then
c 
0
A =0
b
0  abc
0
0
c
Matrices & Determinants
Row(Column) Operations
Following are the notations to evaluate a determinant:
(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.
Similar notations can be used to denote column
operations by replacing R with C.
Matrices & Determinants
Sign System for Expansion of
Determinant
Sign System for order 2 and order 3 are
given by
+ –
+ – +
, – + –
– +
+ – +
Matrices & Determinants
Example-1
Find the value of the following determinants
42
1
6
(i) 28 7 4
14
3
(ii)
2
6
-3 2
2
-1 2
-10
5
2
Solution :
i 
42
28
1
7
6
6×7
4 = 4×7
1
7
6
4
14
3
2
3
2
6 1 6
=7 4 7 4
2×7
 Taking out 7 common
from C1 
2 3 2
= 7×0
=0
 C1 and C3 are identical
Matrices & Determinants
Example –1 (ii)
(ii)
6
-3 2
2
-1 2
-10
5
2
3    2   3 2
 1    2 
5   2 
1 2
5
2
3 3 2
 ( 2) 1 1 2
5
 ( 2)  0
5
Taking out  2 common from C1 
2
 C1 and C2 are identical
0
Matrices & Determinants
Example - 2
1
Evaluate the determinant 1
1
Solution :
1 a b+c
1 a a+b+c
1 b c+a = 1 b a+b+c
1 c
a+b
1 c
1 a 1
=  a+b+c  1 b 1
1 c 1
= a + b + c × 0
a
b
c
b+ c
c+a
a+b
 Applying
c3  c2 +c3 
a+b+c
 Taking  a+b+c  common from C3 
 C1 and C3 are identical
=0
Matrices & Determinants
Applications of Determinants
(Area of a Triangle)
The area of a triangle whose vertices are
(x1, y1 ), (x2 , y2 ) and (x3, y3 ) is given by the expression
x1
Δ=
1
x2
2
x3
y1 1
y2 1
y3 1
1
= [x1 (y2 - y3 ) + x2 (y3 - y1 ) + x3 (y1 - y2 )]
2
Matrices & Determinants
Example
Find the area of a triangle whose
vertices are (-1, 8), (-2, -3) and (3, 2).
Solution :
x1
1
Area of triangle = x2
2
x3
=
=
y1 1
-1 8 1
1
y2 1 = -2 -3 1
2
y3 1
3 2 1
1
-1(-3 - 2)- 8(-2 - 3)+1(-4+ 9)
2
1
5+ 40+5 = 25 sq.units
2
Matrices & Determinants
Condition of Collinearity of
Three Points
If A (x1 , y1 ), B (x2 , y2 ) and C (x3 , y3 ) are three points,
then A, B, C are collinear
 Area of triangle ABC = 0
x1

1
x2
2
x3
y1 1
x1
y1 1
y2 1 = 0  x2
y2 1 = 0
y3 1
y3 1
x3
Matrices & Determinants
Example
If the points (x, -2) , (5, 2), (8, 8) are collinear,
find x , using determinants.
Solution :
Since the given points are collinear.
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
Matrices & Determinants
The Cofactor Matrix
Matrices & Determinants
Finding inverse of nonsingular matrices
Matrices & Determinants
Systems of Linear Equations
Matrices & Determinants
Example:
𝑇ℎ𝑒𝑜𝑟𝑒𝑚 1
𝑇ℎ𝑒𝑜𝑟𝑒𝑚 2
Matrices & Determinants
Solution of System of 2 Linear
Equations (Cramer’s Rule)
Let the system of linear equations be
a1x +b1y = c1
... i
a2 x +b2 y = c2
... ii
Then x =
D1
D
where D =
, y=
D2
D
a1
b1
a2
b2
provided D  0,
, D1 =
c1
b1
c2
b2
and D2 =
a1
c1
a2
c2
Matrices & Determinants
Cramer’s Rule (Con.)
Note :
1
If D  0,
then the system is consistent and has unique solution.
2 
If D = 0 and D1 = D2 = 0,
then the system is consistent and has infinitely many
solutions.
3
If D = 0 and one of D1, D2  0,
then the system is inconsistent and has no solution.
Matrices & Determinants
Example
Using Cramer's rule , solve the following
system of equations 2x-3y=7, 3x+y=5
Solution :
D=
D1 =
D2 =
2 -3
3
1
7 -3
5
1
2 7
3 5
= 2+9 =11  0
=7+15=22
=10-21=-11
D0
By Cramer's Rule x =
D1 22
D
-11
=
= 2 and y = 2 =
= -1
D 11
D
11
Matrices & Determinants
Solution of System of 3 Linear
Equations (Cramer’s Rule)
Let the system of linear equations be
... i
a1x +b1y + c1z = d1
a2 x +b2 y + c2z = d2
... ii
a3 x +b3 y + c3z = d3
... iii
Then x =
D1
D
, y=
D2
D
a1
b1
where D = a2
a3
, z=
D3
c1
D
provided D  0,
d1
b1
b2
c2 , D1 = d2
b2
b3
c3
b3
a1
b1
and D3 = a2
a3
b2
d2
b3
d3
d3
c1
a1
d1
c1
c2 , D2 = a2
c3
a3
d2
c2
d3
c3
d1
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.
(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.
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
5
D1 = 2
-1 4
3 5
-1 -2 6
= 5(18+10) + 1(12-25)+4(-4 -15)
= 140 –13 –76 =140 - 89
= 51  0
= 5(18+10)+1(12+5)+4(-4 +3)
= 140 +17 –4
= 153
Matrices & Determinants
Solution Cont.
5
D2 = 2
5
2
4
5
5 -1 6
5 -1
D3 = 2 3
5
2
5 -2 -1
= 5(12 +5)+5(12 - 25)+ 4(-2 - 10)
= 85 + 65 – 48 = 150 - 48
= 102
= 5(-3 +4)+1(-2 - 10)+5(-4-15)
= 5 – 12 – 95 = 5 - 107
= - 102
D0
 By Cramer's Rule x =
and z =
D1 153
D
102
=
= 3, y = 2 =
=2
D
51
D
51
D3 -102
=
= -2
D
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

We have D = 1
3
1
-2
6
- 1

1  = 1 10 - 6  - 1 -5 - 3  - 1 6 + 6 
- 5 
= 4 + 8 - 12 = 0
 The system has infinitely many solutions.
Putting z = k, in first two equations, we get
x + y = k, x – 2y = -k
Matrices & Determinants
Solution (Con.)
 By Cramer's rule x =
y=
D2
D
=
1
k
1
-k
1
1
1
-2
=
D1
D
=
k
1
-k
-2
1
1
1
-2
=
-2k + k
k
=
-2 - 1
3
-k - k
2k
=
-2 - 1
3
These values of x, y and z = k satisfy (iii) equation.
x=
k
2k
, y=
, z = k , where k  R
3
3
Matrices & Determinants