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**1.**The symbol is called the determinant of order two.

It's value is given by : D = a_{1} b_{2} − a_{2} b_{1}

**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 ; R_{1}, R_{2}, R_{3} or C_{1}, C_{2}, C_{3}.

**3.** Following examples of short hand writing large expressions are :

(i) The lines : a_{1}x + b_{1}y + c_{1} = 0........ (1)

a_{2}x + b_{2}y + c_{2} = 0........ (2)

a_{3}x + b_{3}y + c_{3} = 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 a_{1} in (Key Concept 2) is & the minor of b_{2} 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 = a_{11}M_{11} − a_{12}M_{12} + a_{13}M_{13} OR D = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} & 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 : a_{1}A_{2} + b_{1}B_{2} + c_{1}C_{2} = 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 a_{1}x + b_{1}y + c_{1} = 0 & a_{2}x + b_{2}y + c_{2}_{ }= 0 then :

Given equations are inconsistent & Given equations are dependent

**9. CRAMER'S RULE :** [ SIMULTANEOUS EQUATIONS INVOLVING THREE UNKNOWNS]

Let, a_{1}x + b_{1}y + c_{1}z = d_{1} ...(I) ; a_{2}x + b_{2}y + c_{2}z = d_{2} ... (II) ; a_{3}x + b_{3}y + c_{3}z = d_{3} ... (III)

Then, .

Where

NOTE : (a) If D ≠ 0 and alteast one of D_{1}, D_{2}, D_{3} ≠ 0, then the given system of equations are consistent and have unique non trivial solution.

(b) If D ≠ 0 & D_{1} = D_{2} = D_{3} = 0, then the given system of equations are consistent and have trivial solution only.

(c) If D = D_{1} = D_{2} = D_{3 }= 0, then the given system of equations are consistentand have infinite solutions. In case represents these parallel planes then also D = D_{1} = D_{2} = D_{3} = 0 but the system is inconsistent.

(d) If D = 0 but at least one of D_{1}, D_{2}, D_{3} 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 a_{1}x + b_{1}y + c_{1}z = 0 ; a_{2}x + b_{2}y + c_{2}z = 0 & a_{3}x + b_{3}y + c_{3}z = 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.

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