Types of Determinants and Their Properties JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Types of Determinants and Their Properties JEE Notes | EduRev

The document Types of Determinants and Their Properties JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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F. DETERMINANT

(i) Submatrix : Let A be a given matrix. The matrix obtained by deleting some rows or columns of A is called as submatrix of A.

e.g. Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev are all submatrices of A.

(ii) Determinant of A Square Matrix :

Let A [a]1 x 1 be a 1 x 1 matrix. Determinant A is defined as |A| = a  eg. A = [–3]1 x 1  |A| = –3 

Types of Determinants and Their Properties JEE Notes | EduRev ,then |A| is defined as ad –bc.


(iii)Minors & Cofactors : Let Δ be a determinant. Then minor of element aij, denoted by Mij is defined as the determinant of the submatrix obtained by deleting ith row & jth column of Δ.

Cofactor of element aij, denoted by Cij is defined as Cij = (–1)i + j Mij.

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev


(iv)Determinant : Let A = [aij]n be a square matrix (n > 1). Determinant of A is defined as the sum of products of elements of any one row (or one column) with corresponding cofactors.

Types of Determinants and Their Properties JEE Notes | EduRev

  |A| = a11C11 + a12C12 + a13C13 (using first row) Types of Determinants and Their Properties JEE Notes | EduRev

|A| = a12 C12 + a22C22 + a32C32 (using second column)  Types of Determinants and Their Properties JEE Notes | EduRev

G. PROPERTIES OF DETERMINANTS


P- 1 : The value of a determinant remains unaltered , if the rows & columns are inter changed . e.g. If
Types of Determinants and Their Properties JEE Notes | EduRev  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.

Types of Determinants and Their Properties JEE Notes | EduRev

P-3 : If a determinant has any two rows (or columns) identical , then its value is zero.
Types of Determinants and Their Properties JEE Notes | EduRev


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  Types of Determinants and Their Properties JEE Notes | EduRev


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.


Types of Determinants and Their Properties JEE Notes | EduRev

 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).
Types of Determinants and Their Properties JEE Notes | EduRev

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


Ex.17 Find the value of the determinant
Types of Determinants and Their Properties JEE Notes | EduRev

Sol.

Types of Determinants and Their Properties JEE Notes | EduRev

Ex.18  A is a n × n matrix (n > 2) [aij] where Types of Determinants and Their Properties JEE Notes | EduRev  Find determinant A.

Sol.

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

⇒ value of determinant is zero.

H. MULTIPLICATION OF TWO DETERMINANTS

(i)  Types of Determinants and Their Properties JEE Notes | EduRev

Similarly two determinants of order three are multiplied.

(ii)  Types of Determinants and Their Properties JEE Notes | EduRev  Types of Determinants and Their Properties JEE Notes | EduRev where Ai , Bi , Ci are cofactors

PROOF : Consider   Types of Determinants and Their Properties JEE Notes | EduRev

Note : a1A2 + b1B2 + c1C= 0 etc.

therefore   Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev


Ex.19 Prove that 

Types of Determinants and Their Properties JEE Notes | EduRev = (a – b) (a – c) (a – d) (b – c) (b – d) (c – d).


Sol.

Applying R2 → R2- R1, R3 → R- R1, we get

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

=(a – b) (c – d) (a – c) (b – d)  Types of Determinants and Their Properties JEE Notes | EduRev

= (a – b) (c – d) (a – c) (b – d) [(a + c) (b + d) – (a + b) (c + d)]

= (a – b) (c – d) (a – c) (b – d) (ab +cd – ac – bd) = (a – b) (a – c) (a – d) (b – c) (b – d) (c – d).

Alternatively : 

Types of Determinants and Their Properties JEE Notes | EduRev  and using c3 → c3 + 2 a b c d . c3

Types of Determinants and Their Properties JEE Notes | EduRev

Ex.20 Show that

Types of Determinants and Their Properties JEE Notes | EduRev  (where r2 = x2 + y+ z2 & u2 = x y + y z + z x)

Sol. Consider the determinant ,  Types of Determinants and Their Properties JEE Notes | EduRev We see that the L.H.S. determinant has its constituents
which are the co-factor of Δ. Hence L.H.S. determinant

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

Ex.21 Without expanding, as for as possible, prove that

Types of Determinants and Their Properties JEE Notes | EduRev = (x – y) (y – z) (z – x) (x + y + z)

Sol.

Types of Determinants and Their Properties JEE Notes | EduRev for x = y, D = 0 (since C1 and C2 are identical)

Hence (x – y) is a factor of D (y – z) and (z – x) are factors of D. But D is a homogeneous expression of the 4th degree is x, y, z.
∴ There must be one more factor of the 1st degree in x, y, z say k (x + y + z) where k is a constant.
Let D = k (x – y) (y – z) (z – x) (x + y + z),Putting x = 0, y = 1, z = 2

Types of Determinants and Their Properties JEE Notes | EduRev  = k (0 – 1) (1 – 2) (2 – 0) (0 + 1 + 2)

⇒ L(8 – 2) = k(–1) (–1) (2) (3) ∴ k = 1 ∴ D = (x - y) (y - z) (z - x) (x + y + z)

 

Ex.22 Prove that  Types of Determinants and Their Properties JEE Notes | EduRev


Sol.

Given that Δ =  Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

Ex.23 Express  Types of Determinants and Their Properties JEE Notes | EduRevas product of two determinants.


Sol. The given determinant is 

Types of Determinants and Their Properties JEE Notes | EduRev

Types of Determinants and Their Properties JEE Notes | EduRev

with the help of row–by–row multiplication rule.

Ex.24 

Types of Determinants and Their Properties JEE Notes | EduRevExpress the determinant D as a product of two determinants. Hence or otherwise show that D = 0.

Sol.

Types of Determinants and Their Properties JEE Notes | EduRev as can be seen by applying row–by–row multiplication rule., Hence D = 0.

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