Important Determinant Formulas Formulas for JEE and NEET

1.The symbol  is called the determinant of order two.
It's value is given by  : D = a₁ b₂ − a₂ b₁

2.The symbol   is called the determinant of order three.
Its value can be found as : D =
Or
 and so on. In this manner we can expand a determinant in 6 ways using elements of  ; R₁, R₂, R₃  or  C₁, C₂, C₃.

3. Following examples of short hand writing large expressions are :
(i) The lines : a₁x + b₁y + c₁ = 0........ (1)
a₂x + b₂y + c₂ = 0........ (2)
a₃x + b₃y + c₃ = 0........ (3)
are concurrent if,  Condition for the consistency of  three  simultaneous linear equations  in 2 variables.
(ii) ax^² + 2 hxy + by^² + 2 gx + 2 fy + c = 0 represents a pair of straight lines if abc + 2 fgh − af^² − bg^² − ch^² = 0 =
(iii) Area of a  triangle whose vertices are  (x_r, y_r) ;  r = 1 , 2 , 3  is  :  If  D = 0 then the three points are collinear.
(iv) Equation of a straight line passing through

4. MINORS  : The minor of a given element of a determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands For example, the minor of a₁ in (Key Concept 2)  is &  the minor of b₂ is 

. Hence a determinant of order two will have “4 minors” & a determinant of order  three will have “9 minors” .

5. COFACTOR : If M_ij represents the minor of some typical element then the cofactor is defined as  :
C_ij = (−1)^i+j . M_ij  ;  Where i & j denotes the row & column in which the particular element  lies. Note that the value of a determinant of order three in terms of ‘Minor’ & ‘Cofactor’ can be written as :  D = a₁₁M₁₁ − a₁₂M₁₂ + a₁₃M₁₃ OR  D = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃  & so on .......

6. PROPERTIES  OF  DETERMINANTS : 
P−1 : The value of a determinant remains unaltered, if the rows & columns are inter changed. e.g. if D =  D′ D & D′ are transpose of each other.  If D′ = − D then it is SKEW SYMMETRIC determinant but D′ = D ⇒ 2 D = 0 ⇒ D = 0 ⇒ Skew  symmetric determinant of third order  has the value zero.
P−2 : If any two rows (or  columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g. Let:   Then D′ = − D.
P−3 : If a determinant has any two rows (or columns) identical, then its value is zero. e.g. Let D =  then it can be verified that D = 0.
P−4 : If all the elements of any row (or column) be multiplied by the same number, then the determinant is multiplied by that number.
e.g. If D  Then D′= KD
P−5 : If each element of any row (or column) can be expressed as a sum of two terms then the determinant can be expressed as the sum of two determinants. e.g.
P−6 : 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. Let D  Then  D′ = D.
Note : that while applying this property ATLEAST ONE ROW (OR COLUMN) must remain unchanged.
P−7 : If by putting  x = a the value of a determinant vanishes then (x − a) is a  factor of the determinant.

7. MULTIPLICATION OF TWO DETERMINANTS : (i)  Similarly two determinants of order three are multiplied.
(ii) If D =  where A_i, B_i, C_i are cofactors
PROOF : Consider  Note : a₁A₂ + b₁B₂ + c₁C₂ = 0 etc. therefore,

8. SYSTEM OF LINEAR EQUATION  (IN TWO VARIABLES) :
(i) Consistent Equations : Definite & unique solution.  [intersecting lines]
(ii) Inconsistent Equation : No solution. [Parallel line]
(iii) Dependent equation : Infinite solutions.  [Identical lines]
Let  a₁x + b₁y + c₁ = 0  & a₂x + b₂y + c₂ = 0 then :
 Given equations are inconsistent &  Given equations are dependent

9. CRAMER'S  RULE : [ SIMULTANEOUS EQUATIONS INVOLVING THREE UNKNOWNS]
Let, a₁x + b₁y + c₁z = d₁ ...(I) ; a₂x + b₂y + c₂z = d₂ ... (II) ; a₃x + b₃y + c₃z = d₃ ... (III)
Then, .
Where
NOTE : (a) If  D ≠ 0 and alteast one of D₁, D₂, D₃ ≠ 0, then the given system of equations are consistent and have unique non trivial solution.
(b) If  D ≠ 0  &  D₁ = D₂ = D₃ = 0,  then the given system of equations are consistent and have trivial solution only.
(c) If D = D₁ = D₂ = D₃ = 0, then the given system of equations are consistentand have infinite solutions. In case represents these parallel planes then also D = D₁ = D₂ = D₃ = 0 but the system is inconsistent.
(d) If D = 0 but at least one of D₁, D₂, D₃ is not zero then the equations are in consistent and have no solution.

10. If x, y, z  are not all zero, the condition for a₁x + b₁y + c₁z = 0  ;  a₂x + b₂y + c₂z = 0  & a₃x + b₃y + c₃z = 0 to be consistent in x, y, z is that  Remember that if a  given system of linear equations have Only Zero Solution for all its variables then the given equations are said to have TRIVIAL SOLUTION.