Introducton to Determinants - JEE PDF Download

Introduction to Determinants

The development of determinants took place when mathematicians were trying to solve a system of simultaneous linear equations.

Mathematicians defined the symbol

as a determinant of order 2 and the four numbers arranged in row and column were called its elements. If we write the coefficients of the equations in the following form

, then such an arrangement is called a determinant. In a determinant, horizontal lines are known as rows and vertical lines are known as columns. The shape of every determinant is a square. If a determinant is of order n, then it contains n rows and n columns.

are determinants of second and third order, respectively.
For every square matrix A of order n x n, there exists a number associated with it called the determinant of a square matrix.
For a matrix of 1 x 1, the determinant is A = [a].
For a 2 x 2 matrix,
the determinant is ad - bc

the value of the determinant is = a (ei − fh) − b (di − fg) + c (dh − eg).
Note:

• The number of elements in a determinant of order n is n2.
• A determinant of order 1 is the number itself.

Properties of Determinants

• There will be no change in the value of the determinant if the rows and columns are interchanged.
• Suppose any two rows or columns of a determinant are interchanged, then its sign changes.
• If any two rows or columns of a determinant are the same, then the determinant is 0.
• If any row or column of the determinant is multiplied by a variable k, then its value is multiplied by k.
• Say if some or all elements of a row or column are expressed as the sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.

Illustration 1: Expand

by Sarrus rules.
Solution: By using Sarrus rule, i.e.,
we can expand the given determinant.

Here,
 ⇒ Δ = 15 - 36 + 90 + 16 + 135 + 10 = 230
Illustration 2: Evaluate the determinant :

Solution: By using the determinant expansion formula, we can get the result.
We have

Symmetric and Skew Symmetric Determinants

Symmetric Determinant
A determinant is called a symmetric determinant if a_ij = a_ji, ∀ i, j

Skew Symmetric Determinant
A determinant is called a skew symmetric determinant if a_ij = -a_ji, ∀ i, j for every element.

Note: (i) det |A| = 0 ⇒A is singular matrix (ii) det | A | ≠ 0 ⇒A is non-singular matrix

Multiplication of Two Determinants

(a) Multiplication of two second order determinants is as follows: (as R to C method)

(b) Multiplication of two third order determinants is defined as follows

Note:
(i) The two determinants to be multiplied must be of the same order.
(ii) To get the Tmn (term in the mth row nth column) in the product, Take the mth row of the 1st determinant and multiply it by the corresponding terms of the nth column of the 2nd determinant and add.
(iii) This method is the row-by-column multiplication rule for the product of 2 determinants of the nth order determinant.
(iv) IfΔ′ is the determinant formed by replacing the elements of a Δ of order n with their corresponding co-factors, then
Δ’ = Δn-1 . Δ’ is called the reciprocal determinant.

Illustration 3: Reduce the power of the determinant

Solution: By multiplying the given determinant two times, we get the determinant as required.