Notes: Determinant

 Table of contents Determinants of the Third Order Value of a Determinant Minors and Cofactors Expansion of a Determinant in terms of the Elements of any Row or Column Properties of Determinants Multiplication of Determinants: Special Determinants: System of Linear Equations (Cramer's Rule) :

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 Mij represents the minor of the element belonging to ith row and jth column then the cofactor of that element is given by : Cij = (–1)i + j . Mij.

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

The sum of the products of the elements of any row (or column) of a matrix with their corresponding cofactors always equals the value of the determinant of the matrix.

This determinant $DD$ can be expressed in any of the following six forms:

1. $a1A1+b1B1+c1C1a_1 A_1 + b_1 B_1 + c_1 C_1$
2. $a1A1+a2A2+a3A3a_1 A_1 + a_2 A_2 + a_3 A_3$
3. $a2A2+b2B2+c2C2a_2 A_2 + b_2 B_2 + c_2 C_2$
4. $b1B1+b2B2+b3B3$
5. $a3A3+b3B3+c3C3a_3 A_3 + b_3 B_3 + c_3 C_3$
6. $c1C1+c2C2+c3C3c_1 C_1 + c_2 C_2 + c_3 C_3$

Here, Ai, Bi, and $C_i$Ci (for $i=1,2,3i = 1, 2, 3$) denote the cofactors of ai, $b_i$bi, and $c_i$ci respectively.

Additionally, the sum of the products of the elements of any row (or column) with the cofactors of another row (or column) is always equal to zero. Consequently, we have:

$a2A1+b2B1+c2C1=0a_2 A_1 + b_2 B_1 + c_2 C_1 = 0$,  $b1A1+b2A2+b3A3=0b_1 A_1 + b_2 A_2 + b_3 A_3 = 0$  and so forth.

In these equations, $A_i$Ai, Bi, and Ci (for $i=1,2,3i = 1, 2, 3$) are again the cofactors of ai, $b_i$bi, and $c_i$ci respectively.

## Properties of Determinants

(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 (Ci ) operation of the form Ci + aCj + bCk (j, k  is not equal to i ) or row (Ri ) operation of the form Ri  →  Ri + aRj + bRk (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 = Dn–1

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

## 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:

The document Notes: Determinant | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## FAQs on Notes: Determinant - Mathematics (Maths) for JEE Main & Advanced

 1. How do you find the value of a determinant?
Ans. The value of a determinant is found by multiplying the elements of a row (or column) by their respective minors and adding them up.
 2. What are minors and cofactors of a determinant?
Ans. Minors of a determinant are the determinants formed by removing the row and column of a specific element. Cofactors are the minors multiplied by -1 raised to the power of the sum of the row and column numbers.
 3. How can a determinant be expanded in terms of the elements of any row or column?
Ans. A determinant can be expanded by choosing a row or column and multiplying each element by its cofactor, then summing them up to get the value of the determinant.
 4. What are some properties of determinants?
Ans. Some properties of determinants include linearity, scalar multiplication, the determinant of the identity matrix, the determinant of a transpose, and the determinant of a product of matrices.
 5. How does Cramer's Rule relate to determinants in solving a system of linear equations?
Ans. Cramer's Rule uses determinants to solve a system of linear equations by expressing the solutions in terms of ratios of determinants.

## Mathematics (Maths) for JEE Main & Advanced

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