## Determinant

**1.** The symbol is called the determinant of order two. Its value is given by : D = a_{1} b_{2} − a_{2} b_{1}

**2.** The symbol s called the determinant of order three .

Its value can be found as:

OR

.

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)

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 (x_{r}, y_{r}) ; 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 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_{1}_{1 }− a_{12}M_{12} + a_{13}M_{13} OR D = a_{1}_{1}C_{1}_{1} + a_{12}C_{12} + a_{13}C_{13} & 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.

e.g.

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 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:

&

**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 atleast one of D

_{1}, D

_{2}, D

_{3}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 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.