DETERMINANT
1. MINORS :
The minor of a given element of determinant is the determinant of
the elements which remain after deleting the row & the column in
which the given element stands.
a1 b1 c1
b c2
For example, the minor of a1 in a2 b2 c2 is 2 & the
b3 c3
a 3 b3 c 3
a1 c1
minor of b2 is .
a3 c3
Hence a determinant of order three will have “ 9 minors”.
2. COFACTORS :
If Mij represents the minor of the element belonging to ith row and jth
column then the cofactor of that element : Cij = (–1)i + j. Mij ;
Important Note :
a1 a2 a3
Consider D= b1 b2 b3
c1 c2 c3
Let A1 be cofactor of a1, B2 be cofactor of b2 and so on, then,
(i) a1A1 + b1B1 + c1C1 = a1A1 + a2A2 + a3A3 = .............. = D
(ii) a2A1 + b2B1 + c2C1 = b1A1 + b2A2 + b3A3 = ............. = 0
3. PROPERTIES OF DETERMINANTS:
(a) The value of a determinants remains unaltered, if the rows &
corresponding columns are interchanged.
(b) If any two rows (or columns) of a determinant be interchanged,
the value of determinant is changed in sign only. e.g.
a1 b1 c1 a 2 b2 c2
Let D = a 2 b2 c2 & D ' = a1 b1 c1 Then D' = – D.
a 3 b3 c3 a3 b3 c3
(c) If a determinant has any two rows (or columns) identical or in
same proportion, then its value is zero.
(d) If all the elements of any row (or column) be multiplied by the
same number, then the determinant is multiplied by that number.
a1 + x b1 + y c1 + z a1 b1 c1 x y z
(e) a2 b2 c2 = a2 b2 c2 + a 2 b2 c2
a3 b3 c3 a3 b3 c3 a3 b3 c3
(f) The value of a determinant is not altered by adding to the
elements of any row (or column ) the same multiples of the
corresponding elements of any other row (or column) e.g.
a1 b1 c1
Let D= a2 b2 c2
a3 b3 c3
a1 + ma2 b1 + mb2 c1 + mc2
D' = a2 b2 c2 . Then D' = D.
a3 + na1 b3 + nb1 c 3 + nc1
Note : While applying this property ATLEAST ONE ROW
(OR COLUMN) must remain unchanged.
(g) If the elements of a determinant D are rational function of x
and two rows (or columns) become identical when x = a, then x
– a is a factor of D.
Again, if r rows become identical when a is substituted for x,
then (x – a)r–1 is a factor of D.
f1 f2 f3
(h) If D(x) = g1 g2 g3 , where fr, gr, hr; r = 1, 2, 3 are three
h1 h2 h3
differentiable functions.
f '1 f '2 f '3 f1 f2 f3 f1 f2 f3
d
then D(x) = g1 g2 g3 + g '1 g '2 g '3 + g1 g2 g3
dx
h1 h2 h3 h1 h2 h3 h '1 h '2 h '3
4. MULTIPLICATION OF TWO DETERMINANTS :
a1 b1 l m1 a1 l1 + b1 l2 a1 m1 + b1 m2
´ 1 =
a 2 b2 l2 m2 a2 l1 + b2 l2 a2 m1 + b2 m2
Similarly two determinants of order three are multiplied.
(a) Here we have multiplied row by column. We can also multiply
row by row, column by row and column by column.
(b) If D' is the determinant formed by replacing the elements of
determinant D of order n by their corresponding cofactors then
D' = Dn–1
5. SPECIAL DETERMINANTS :
(a) Symmetric Determinant :
Elements of a determinant are such that aij = aji.
a h g
e.g. h b f = abc +2fgh - af 2 - bg2 - ch2
g f c
(b) Skew Symmetric Determinant :
If aij = –aji then the determinant is said to be a skew symmetric
determinant. Here all the principal diagonal elements are zero.
The value of a skew symmetric determinant of odd order is zero
and of even order is perfect square.
0 b -c
e.g. -b 0 a = 0
c -a 0
(c) Other Important Determinants :
1 1 1 1 1 1
(i) a b c = a b c = (a - b)(b - c)(c - a)
bc ac ab a 2 b2 c 2
a b c
(ii) b c a = - (a3 + b3 + c 3 - 3abc)
c a b
6. SYSTEM OF EQUATION :
(a) System of equation involving two variable :
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
Consistent Inconsistent
(System of equation has solution) (System of equation
has no solution)
a1 b1 c1
unique solution Infinite solution = ¹
a1 b1 a1 b1 c1 a2 b2 c2
¹ = = (Equations represents
a 2 b2 a 2 b2 c2
or a1b2 – a 2b1¹ 0 (Equations represents parallel disjoint lines)
(Equations represents coincident lines)
intersecting lines)
b1 c1 c1 a1 a1 b1 D1 D
If D1 = , D2 = , D= , then x = , y= 2
b2 c2 c2 a2 a2 b2 D D
(b) System of equations involving three variables :
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
To solve this system we first define following determinants
a1 b1 c1 d1 b1 c1
D = a2 b2 c2 , D1 = d2 b2 c2 ,
a3 b3 c3 d3 b3 c3
a1 d1 c1 a1 b1 d1
D2 = a 2 d2 c2 , D3 = a 2 b2 d2
a3 d3 c3 a3 b3 d3
Now following algorithm is used to solve the system.
Check value of D
D¹ 0 D=0
Consistent system Check the values of
and has unique solution D1, D2 and D3
D D D
x = 1, y = 2,z = 3
D D D
Atleast one of D1, D2 and D3 D1= D2 = D3 = 0
is not zero
Put z = t and solve any
Inconsistent system two equations to get the
values of x & y in terms of t
If these values of x, y and z in The values of x, y, z doesn't
terms of t satisfy third equation satisfy third equation
Consistent system Inconsistent system
(System has infinite solutions)
Note :
(i) Trivial solution : In the solution set of system of equation if all
the variables assumes zero, then such a solution set is called
Trivial solution otherwise the solution is called non-trivial solution.
(ii) If d1 = d2 = d3 = 0 then system of linear equation is known as
system of Homogeneous linear equation which always posses
atleast one solution (0, 0, 0).
(iii) If system of homogeneous linear equation posses non-zero/non-
trivial solution then D = 0.
In such case given system has infinite solutions.
