1. The symbol is called the determinant of order two. Its value is given by : D = a1 b2 − a2 b1
2. The symbol s called the determinant of order three .
Its value can be found as:
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)
Condition for the consistency of three simultaneous linear equations in 2 variables.
(ii) ax² + 2 hxy + by² + 2 gx + 2 fy + c = 0 represents a pair of straight lines if
(iii) Area of a triangle whose vertices are (xr, yr) ; r = 1 , 2 , 3 is :
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,
Hence a determinant of order two will have “4 minors” & a determinant of order three will have “9 minors” .
5. Cofactor: If Mij represents the minor of some typical element then the cofactor is defined as: Cij = (−1)i+j . Mij ; 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:
- Property 1: The value of a determinant remains unaltered , if the rows & columns are inter changed . e.g.
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.
- Property 2: If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g.
Then D′ = − D.
- Property 3: If a determinant has any two rows (or columns) identical , then its value is zero . e. g.
then it can be verified that D=0
- Property 4: If all the elements of any row (or column) be multiplied by the same number , then the determinant is multiplied by that number.
Then D′ = KD
- Property 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.
- Property 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.
Then D′ = D.
Note: that while applying this property Atleast One Row (Or Column) must remain unchanged.
- Property 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:
Similarly two determinants of order three are multiplied.
where Ai, Bi, Ci are cofactors
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:
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)
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 atleast one of D1 , D2 , D3 is not zero then the equations are inco ns istent 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.