Notes: Determinant | Mathematics (Maths) for JEE Main & Advanced PDF Download

If the equations a₁x + b₁ = 0 and a₂x + b₂ = 0 are satisfied by the same value of x, then a₁b₂ – a₂b₁ = 0.

The expression a₁b₂ – a₂b₁ is called a determinant of the second order, and is denoted by:

A determinant of the second order consists of two rows and two columns.

Determinants of the Third Order

Consider the system of equations:

• a₁x + b₁y + c₁ = 0
• a₂x + b₂y + c₂ = 0
• a₃x + b₃y + c₃ = 0

If these equations are satisfied by the same values of x and y, then on eliminating x and y we get:

a₁(b₂c₃ – b₃c₂) + b₁(c₂a₃ – c₃a₂) + c₁(a₂b₃ – a₃b₂) = 0

The expression on the left is called a determinant of the third order, and is denoted by:

A determinant of the third order consists of three rows and three columns.

Value of a Determinant

The value of a determinant of order three is given by:

D = a₁(b₂c₃ – b₃c₂) – b₁(a₂c₃ – a₃c₂) + c₁(a₂b₃ – a₃b₂)    

Using the Sarrus diagram to find the value of a third-order determinant:

Example 1: Find the value: 

Solution:

= (27 + 42) – 2 (–36 –12) + 3 (28 – 6) = 231

Alternatively, Using Sarrus Diagram,

= (27 + 24 + 84) – (18 – 42 – 72)= 135 – (18 – 114) = 231  

Minors and Cofactors

The minor of a given element of a determinant is the determinant obtained by deleting the row and the column in which the given element stands.

For example, the minor of a₁ in

Hence a determinant of order three will have “9 minors”. If M_ij represents the minor of the element belonging to i^th row and j^th column then the cofactor of that element is given by : C_ij = (–1)^{i + j} . M_ij.

Expansion of a Determinant in terms of the Elements of any Row or Column

(a) The value of a determinant remains unaltered, if the rows & columns are inter-changed.

(b) If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g. 

(c) If all the elements of a row (or column) are zero, then the value of the determinant is zero. 

(d) If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.

(e) If all the elements of a row (or column) are proportional (or identical) to the element of any other row, then the determinant vanishes, i.e. its value is zero. 

Example: Prove that:  

(f) If each element of any row (or column) is expressed as a sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants. 

(g) Row - column operation : The value of a determinant remains unaltered under a column (C_i ) operation of the form C_i + aC_j + bC_k (j, k  is not equal to i ) or row (R_i ) operation of the form R_i  →  R_i + aR_j + bR_k (j, k ≠  i). In other words, the value of a determinant is not altered by adding the elements of any row (or column) to the same multiples of the corresponding elements of any other row (or column).

(h) Factor theorem : If the elements of a determinant D are rational integral functions of x and two rows (or columns) become identical when x = a then (x – a) is a factor of D. Note that if r rows become identical when a is substituted for x, then (x – a)^r–1 is a factor of D.

Multiplication of Determinants:

Similarly two determinants of order three are multiplied. (a) Here we have multiplied row by column. We can also multiply row by row, column by row and column by column. (b) If D1 is the determinant formed by replacing the elements of determinant D of order n by their corresponding cofactors then D₁ = D^n–1

Example: Let a & b be the roots of equation ax² + bx + c = 0 and S_n = aⁿ + bⁿ for n ≥  1. Evaluate the value of the determinant

Special Determinants:

System of Linear Equations (Cramer's Rule) :

(a) Equations involving two variables : 

(i) Consistent Equations : Definite & unique solution (Intersecting lines) 

(ii) Inconsistent Equations : No solution (Parallel lines) 

(iii) Dependent Equations : Infinite solutions (Identical lines)

(b) Equations Involving Three variables : 

Note : 

(i) If D ≠  0 and atleast one of D₁ , D₂ , D₃ ≠  0, then the given system of equations is consistent and has unique non trivial solution. 

(ii) If D ≠  0 & D₁ = D₂ = D₃ = 0, then the given system of equations is consistent and has trivial solution only. 

(iii) If D = 0 but atleast one of D₁ , D₂ , D₃ is not zero then the equations are inconsistent and have no solution. 

(iv) If D = D₁ = D₂ = D₃ = 0, then the given system of equations may have infinite or no solution.

(c) Homogeneous system of linear equations :

 Applications of Determinants: