Important Determinants Formulas for JEE and NEET

 Page 1


  
PROBLEM-SOLVING TACTICS
Let 
1 2 3
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
? = , then 
1 2 3
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
' ' '
' ?= and in general
n n n
1 2 3
n
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
? =
 
where n is any positive integer and 
n
f (x) denotes the 
th
n derivative of f(x).
Let 
f(x) g(x) h(x)
(x) a b c
mn
? =
l
, where a, b, c, l, m and n are constants.
? 
b b b
a a a b
a
f(x)dx g(x)dx h(x)dx
(x)dx a b c
m n
? =
? ? ?
?
l
If the elements of more than one column or rows are functions of x then the integration can be done only 
after evaluation/expansion of the determinant.
FORMULAE SHEET
(a) Determinant of order 3 × 3 =
1 1 1
2 2 2
3 3 3
ab c
a b c
ab c
 = 
2 2 2 2 2 2
1 1 1
3 3 3 3 3 3
b c a c a b
a b c
b c a c bb
- +
(b) In the determinant D = 
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
, minor of 
12
a is denoted as 
21 23
12
31 33
a a
M
a a
= and so on.
(c) Cofactor of an element 
ij
a =
i j
ij ij
C ( 1) M
+
= - 
Page 2


  
PROBLEM-SOLVING TACTICS
Let 
1 2 3
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
? = , then 
1 2 3
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
' ' '
' ?= and in general
n n n
1 2 3
n
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
? =
 
where n is any positive integer and 
n
f (x) denotes the 
th
n derivative of f(x).
Let 
f(x) g(x) h(x)
(x) a b c
mn
? =
l
, where a, b, c, l, m and n are constants.
? 
b b b
a a a b
a
f(x)dx g(x)dx h(x)dx
(x)dx a b c
m n
? =
? ? ?
?
l
If the elements of more than one column or rows are functions of x then the integration can be done only 
after evaluation/expansion of the determinant.
FORMULAE SHEET
(a) Determinant of order 3 × 3 =
1 1 1
2 2 2
3 3 3
ab c
a b c
ab c
 = 
2 2 2 2 2 2
1 1 1
3 3 3 3 3 3
b c a c a b
a b c
b c a c bb
- +
(b) In the determinant D = 
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
, minor of 
12
a is denoted as 
21 23
12
31 33
a a
M
a a
= and so on.
(c) Cofactor of an element 
ij
a =
i j
ij ij
C ( 1) M
+
= - 
17.20  |  Determinants
(d) Properties of determinants: 
(i) Reflection property: 
i j ji
A A
× ×
= 
(ii) All-zero property: If all the elements of a row (or column) are zero, then the determinant is zero.
(iii) Proportionality (Repetition) Property: If all the elements of a row (or column) are proportional 
(identical) to the elements of some other row (or column), then the determinant is zero.
(iv) Switching Property: The interchange of any two rows (or columns) of the determinant changes its sign.
(v) Scalar Multiple Property: If all the elements of a row (or column) of a determinant are multiplied by a 
non-zero constant, then the determinant gets multiplied by the same constant.
(vi) Sum Property: 
1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3
a b cd a cd b cd
a b cd a cd b cd
a b cd a cd b cd
+
+ = +
+
(vii) Property of Invariance: 
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 bc a b c bc
a bc a b c bc
a bc a b c bc
+ a +ß
= + a +ß
+ a +ß
  That is, a determinant remains unaltered under an operation of the form 
i i j k
CC C C ? + a +ß , where j,k i ? , 
or an operation of the form 
i i j k
RR R R ? + a +ß ,  where j,k i ? .
(viii) Triangle Property: 
1 2 3 1
2 3 2 2 1 23
3 3 3 3
a a a a 00
0 b b a b 0 ab c
0 0c a b c
= =
(e) Cramer’s rule : if 
1 1 1 1
a x b y c z d + + = , 
2 2 2 2
ax b y c z d + + = and 
3 3 3 3
ax b y c z d + + = then 
1
x
?
=
?
,
2
y
?
=
?
,
3
z
?
=
?
 
where 
 
1 1 1
2 2 2
3 3 3
ab c
a b c
ab c
?= , 
1 1 1
1 2 2 2
3 3 3
d b c
d b c
d b c
?= , 
1 1 1
2 2 2 2
3 3 3
a d c
a dc
a d c
? = and 
1 1 1
3 2 2 2
3 3 3
ab d
a b d
ab d
?= . 
 And if 
1 1 1
a x b y c 0 + += and 
2 2 2
ax b y c 0 + + = then 
1
x
?
=
?
2
y
?
=
?
.
 Where 
1 1
1
2 2
bc
bc
?= , 
1 1
2
2 2
ca
ca
? = and 
1 1
2 2
ab
a b
?=
(f) (i) lines 
1 1 1
a x b y c 0 + += ,
2 2 2
ax b y c 0 + + = and 
3 3 3
ax b y c 0 + += are concurrent if,  
1 1 1
2 2 2
3 3 3
ab c
a b c 0
ab c
=
 (ii) 
2 2
ax 2hxy by 2gx 2fy c 0 + + + + += represents a pair of straight lines if 
a hg
h b f
gf c
=0
 (iii) area of a triangle whose vertices are 
r r
(x , y ) ; r = 1, 2, 3 is : D = 
1 1
2 2
3 3
x y1
1
x y 1
2
x y1
 (iv) Equation of a straight line passing through 
11
(x , y ) & 
2 2
(x , y ) is 
1 1
2 2
x y 1
x y 10
x y 1
=
Page 3


  
PROBLEM-SOLVING TACTICS
Let 
1 2 3
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
? = , then 
1 2 3
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
' ' '
' ?= and in general
n n n
1 2 3
n
1 2 3
1 2 3
f (x) f (x) f (x)
(x) b b b
c c c
? =
 
where n is any positive integer and 
n
f (x) denotes the 
th
n derivative of f(x).
Let 
f(x) g(x) h(x)
(x) a b c
mn
? =
l
, where a, b, c, l, m and n are constants.
? 
b b b
a a a b
a
f(x)dx g(x)dx h(x)dx
(x)dx a b c
m n
? =
? ? ?
?
l
If the elements of more than one column or rows are functions of x then the integration can be done only 
after evaluation/expansion of the determinant.
FORMULAE SHEET
(a) Determinant of order 3 × 3 =
1 1 1
2 2 2
3 3 3
ab c
a b c
ab c
 = 
2 2 2 2 2 2
1 1 1
3 3 3 3 3 3
b c a c a b
a b c
b c a c bb
- +
(b) In the determinant D = 
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
, minor of 
12
a is denoted as 
21 23
12
31 33
a a
M
a a
= and so on.
(c) Cofactor of an element 
ij
a =
i j
ij ij
C ( 1) M
+
= - 
17.20  |  Determinants
(d) Properties of determinants: 
(i) Reflection property: 
i j ji
A A
× ×
= 
(ii) All-zero property: If all the elements of a row (or column) are zero, then the determinant is zero.
(iii) Proportionality (Repetition) Property: If all the elements of a row (or column) are proportional 
(identical) to the elements of some other row (or column), then the determinant is zero.
(iv) Switching Property: The interchange of any two rows (or columns) of the determinant changes its sign.
(v) Scalar Multiple Property: If all the elements of a row (or column) of a determinant are multiplied by a 
non-zero constant, then the determinant gets multiplied by the same constant.
(vi) Sum Property: 
1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3
a b cd a cd b cd
a b cd a cd b cd
a b cd a cd b cd
+
+ = +
+
(vii) Property of Invariance: 
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 bc a b c bc
a bc a b c bc
a bc a b c bc
+ a +ß
= + a +ß
+ a +ß
  That is, a determinant remains unaltered under an operation of the form 
i i j k
CC C C ? + a +ß , where j,k i ? , 
or an operation of the form 
i i j k
RR R R ? + a +ß ,  where j,k i ? .
(viii) Triangle Property: 
1 2 3 1
2 3 2 2 1 23
3 3 3 3
a a a a 00
0 b b a b 0 ab c
0 0c a b c
= =
(e) Cramer’s rule : if 
1 1 1 1
a x b y c z d + + = , 
2 2 2 2
ax b y c z d + + = and 
3 3 3 3
ax b y c z d + + = then 
1
x
?
=
?
,
2
y
?
=
?
,
3
z
?
=
?
 
where 
 
1 1 1
2 2 2
3 3 3
ab c
a b c
ab c
?= , 
1 1 1
1 2 2 2
3 3 3
d b c
d b c
d b c
?= , 
1 1 1
2 2 2 2
3 3 3
a d c
a dc
a d c
? = and 
1 1 1
3 2 2 2
3 3 3
ab d
a b d
ab d
?= . 
 And if 
1 1 1
a x b y c 0 + += and 
2 2 2
ax b y c 0 + + = then 
1
x
?
=
?
2
y
?
=
?
.
 Where 
1 1
1
2 2
bc
bc
?= , 
1 1
2
2 2
ca
ca
? = and 
1 1
2 2
ab
a b
?=
(f) (i) lines 
1 1 1
a x b y c 0 + += ,
2 2 2
ax b y c 0 + + = and 
3 3 3
ax b y c 0 + += are concurrent if,  
1 1 1
2 2 2
3 3 3
ab c
a b c 0
ab c
=
 (ii) 
2 2
ax 2hxy by 2gx 2fy c 0 + + + + += represents a pair of straight lines if 
a hg
h b f
gf c
=0
 (iii) area of a triangle whose vertices are 
r r
(x , y ) ; r = 1, 2, 3 is : D = 
1 1
2 2
3 3
x y1
1
x y 1
2
x y1
 (iv) Equation of a straight line passing through 
11
(x , y ) & 
2 2
(x , y ) is 
1 1
2 2
x y 1
x y 10
x y 1
=
(g) If 
1 1
2 2
f (x) g (x)
(x)
f (x) g (x)
? = then
1 1
1 1
2 2 2 2
f (x) g (x)
f (x) g (x)
(x)
f (x) g (x) f (x) g (x)
' '
' ?= +
' '
 or 
1 1 1 1
2 2 2 2
f (x) g (x) f (x) g (x)
f (x) g (x) f (x) g (x)
' '
+
' '
(h) If 
f(x) g(x) h(x)
(x) a b c ? =
a ß ?
 then 
f(x)dx g(x)dx h(x)dx
(x)dx a b c ? =
a ß ?
? ? ?
?