Important Formulas: Determinant Formulas

# 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) ax² + 2 hxy + by² + 2 gx + 2 fy + c = 0 represents a pair of straight lines if abc + 2 fgh − af² − bg² − ch² = 0 =
(iii) Area of a  triangle whose vertices are  (xr, yr) ;  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 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 =  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 Ai, Bi, Ci 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 = D= 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.

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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## 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.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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