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Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE PDF Download

Determinant

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

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE 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 | Mathematics (Maths) Class 12 - JEE

then |A| is defined as ad –bc.

(iii) Minors & Cofactors : Let Δ be a determinant. Then the 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 | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

(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 | Mathematics (Maths) Class 12 - JEE

  |A| = a11C11 + a12C12 + a13C13 (using first row)

 Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

|A| = a12 C12 + a22C22 + a32C32 (using second column)  

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Properties of Determinants

P- 1 : The value of a determinant remains unaltered , if the rows & columns are interchanged . e.g. If

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

are transpose of each other.

If D' = - D then it is a 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 are interchanged , the value of determinant is changed in sign only . e.g.

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

P-4 : If all the elements of any row (or column) are multiplied by the same number then the determinant is multiplied by that number.
Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

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 | Mathematics (Maths) Class 12 - JEE

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 | Mathematics (Maths) Class 12 - JEE

Note that while applying this property at least 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
Example: Find the value of the determinant

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Sol.

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE


Example:  A is a n × n matrix (n > 2) [aij] where  Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE 

Find determinant A.
Sol.

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

⇒ value of determinant is zero.

Multiplication of Two Determinants

(i)Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Similarly two determinants of order three are multiplied.

(ii)
Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

where Ai , Bi , Ci are cofactors

PROOF : Consider 

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

 Note : a1A2 + b1B2 + c1C2 = 0 etc.
therefore

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Example:  Prove that

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

= (a – b) (a – c) (a – d) (b – c) (b – d) (c – d)

Sol.
Applying R2 → R2- R1, R3 → R3 - R1, we get

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

= (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 | Mathematics (Maths) Class 12 - JEE

and using c3 → c3 + 2 a b c d . c3

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Example:  Show that

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

(where r2 = x2 + y2 + z2 & u2 = x y + y z + z x)

Sol. Consider the determinant:

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE
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 | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Example:  Without expanding, as for as possible, prove that

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

= (x – y) (y – z) (z – x) (x + y + z)

Sol.
Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

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, then:

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

= 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)

Example: Prove that

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Sol.

Given that Δ = 

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Example: Express

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

as product of two determinants.
Sol. The given determinant is
Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE
with the help of row–by–row multiplication rule.

Example:

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE

Express the determinant D as a product of two determinants. Hence or otherwise show that D = 0.

Sol.

Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE
as can be seen by applying row–by–row multiplication rule., Hence D = 0.

The document Types of Determinants and Their Properties | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Types of Determinants and Their Properties - Mathematics (Maths) Class 12 - JEE

1. What are the different types of determinants?
Ans. There are mainly three types of determinants: 1. Square Determinants: These are the determinants of square matrices, where the number of rows is equal to the number of columns. 2. Row Determinants: These are the determinants obtained by expanding along a row of a square matrix. 3. Column Determinants: These are the determinants obtained by expanding along a column of a square matrix.
2. What are the properties of determinants?
Ans. The properties of determinants include: 1. Multiplicative Property: If a matrix A has a determinant D, then multiplying each element of A by a constant k multiplies the determinant by the same constant, i.e., kD. 2. Row or Column Interchange Property: Interchanging any two rows or columns of a matrix changes the sign of the determinant. 3. Additive Property: If two matrices A and B are of the same order, then the determinant of their sum A + B is equal to the sum of their determinants. 4. Scalar Multiplication Property: If each element of a row or column of a matrix is multiplied by a scalar k, then the determinant of the new matrix is equal to k times the determinant of the original matrix. 5. Zero Row or Column Property: If a matrix has a row or column consisting entirely of zeros, then its determinant is zero.
3. How do determinants help in solving systems of linear equations?
Ans. Determinants play a crucial role in solving systems of linear equations. The determinant of the coefficient matrix can determine whether a system has a unique solution, no solution, or infinitely many solutions. If the determinant of the coefficient matrix is nonzero, then the system has a unique solution. If the determinant is zero, then the system may have either no solution or infinitely many solutions, and further analysis is required using other methods.
4. Can determinants be negative?
Ans. Yes, determinants can be negative. The sign of a determinant depends on the number of row or column interchanges required to transform the matrix to its upper triangular form. If an odd number of row/column interchanges are performed, the determinant will be negative. If an even number of row/column interchanges are performed, the determinant will be positive.
5. What is the significance of determinant value being zero?
Ans. When the determinant of a matrix is zero, it indicates that the matrix is singular or not invertible. In the context of solving systems of linear equations, a determinant of zero implies that the system either has no solution or infinitely many solutions. The determinant being zero suggests a dependency among the equations, leading to an inconsistent or underdetermined system.
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