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Important Determinant Formulas Formulas for JEE and NEET

1.The symbol Important Determinant Formulas Formulas for JEE and NEET is called the determinant of order two.
It's value is given by  : D = a1 b2 − a2 b1

2.The symbol Important Determinant Formulas Formulas for JEE and NEET  is called the determinant of order three.
Its value can be found as : D = Important Determinant Formulas Formulas for JEE and NEET
Or
Important Determinant Formulas Formulas for JEE and NEET 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, Important Determinant Formulas Formulas for JEE and NEET 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 abc + 2 fgh − af² − bg² − ch² = 0 = Important Determinant Formulas Formulas for JEE and NEET
(iii) Area of a  triangle whose vertices are  (xr, yr) ;  r = 1 , 2 , 3  is  : Important Determinant Formulas Formulas for JEE and NEET If  D = 0 then the three points are collinear.
(iv) Equation of a straight line passing through Important Determinant Formulas Formulas for JEE and NEET

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 Important Determinant Formulas Formulas for JEE and NEET&  the minor of b2 is 

Important Determinant Formulas Formulas for JEE and NEET. 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 : 
P−1 : The value of a determinant remains unaltered, if the rows & columns are inter changed. e.g. if D = Important Determinant Formulas Formulas for JEE and NEET 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:  Important Determinant Formulas Formulas for JEE and NEET Then D′ = − D.
P−3 : If a determinant has any two rows (or columns) identical, then its value is zero. e.g. Let D = Important Determinant Formulas Formulas for JEE and NEET 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 Important Determinant Formulas Formulas for JEE and NEET 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. Important Determinant Formulas Formulas for JEE and NEET
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 Important Determinant Formulas Formulas for JEE and NEET 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) Important Determinant Formulas Formulas for JEE and NEET Similarly two determinants of order three are multiplied.
(ii) If D = Important Determinant Formulas Formulas for JEE and NEET where Ai, Bi, Ci are cofactors
PROOF : Consider Important Determinant Formulas Formulas for JEE and NEET Note : a1A2 + b1B2 + c1C2 = 0 etc. therefore, Important Determinant Formulas Formulas for JEE and NEET

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 :
Important Determinant Formulas Formulas for JEE and NEET Given equations are inconsistent & Important Determinant Formulas Formulas for JEE and NEET 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, Important Determinant Formulas Formulas for JEE and NEET.
Where Important Determinant Formulas Formulas for JEE and NEET
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 = D= 0, then the given system of equations are consistentand have infinite solutions. In case Important Determinant Formulas Formulas for JEE and NEETrepresents 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 Important Determinant Formulas Formulas for JEE and NEET 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.

The document Important Determinant Formulas Formulas for JEE and NEET is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on Important Determinant Formulas Formulas for JEE and NEET

1. What is the formula for calculating the determinant of a 2x2 matrix?
Ans. The formula for calculating the determinant of a 2x2 matrix is ad-bc, where a, b, c, and d are the elements of the matrix.
2. How can I find the determinant of a 3x3 matrix using the cofactor expansion method?
Ans. To find the determinant of a 3x3 matrix using the cofactor expansion method, choose any row or column and multiply each element of that row or column by its corresponding cofactor. Then, add up all the products to get the determinant.
3. Is the determinant of a matrix affected by the row or column operations performed on it?
Ans. Yes, the determinant of a matrix changes if we perform row or column operations on it. However, if we perform elementary operations (i.e., multiplying a row or column by a constant, adding one row or column to another, or interchanging two rows or columns), we can use the following rules to find the new determinant: If we multiply a row or column by a constant k, the determinant is multiplied by k. If we add one row or column to another, the determinant remains the same. If we interchange two rows or columns, the determinant is multiplied by -1.
4. What is the relationship between the determinant and the invertibility of a matrix?
Ans. A matrix is invertible if and only if its determinant is nonzero. In other words, if the determinant of a matrix is zero, then the matrix is not invertible.
5. Can the determinant of a matrix be negative?
Ans. Yes, the determinant of a matrix can be negative. In fact, the determinant of a matrix can be positive, negative, or zero, depending on the values of the matrix's elements.
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