Determinants - Introduction JEE Notes | EduRev

JEE : Determinants - Introduction JEE Notes | EduRev

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Definition

To every square matrix A = [aij] of order n, we can associate a number (real or complex) called the determinant of the matrix A, written as det A, where aij is the (i, j)th element of A.

If Determinants - Introduction JEE Notes | EduRev, then the determinant of A, denoted by |A| (or det A), is given by
Determinants - Introduction JEE Notes | EduRev

Note:
(i) Only square matrices have determinants.
(ii) For a matrix A, A is read as the determinant of A and not, as a modulus of A. 

1. Determinant of a matrix of order one: 
Let A = [a] be the matrix of order 1.
⇒ Determinant of A = a

2. Determinant of a matrix of order two:
Let Determinants - Introduction JEE Notes | EduRevbe the matrix of order 2.
⇒ Determinant of A = ad-bc

3. Determinant of a matrix of order three:

  • The determinant of a matrix of order three can be determined by expressing it in terms of second-order determinants which are known as the expansion of a determinant along a row (or a column).
  • Consider the determinant of a square matrix A = [aij] 3×3,
    i.e.Determinants - Introduction JEE Notes | EduRev
    Determinants - Introduction JEE Notes | EduRev

Note:  In general, if A = kB, where A and B are square matrices of order n,
then |A| = kn|B|, where n = 1, 2, 3.

  • We can expand the determinant through any row or column.
    i.e. Δ = a11A11 + a12A12 + a13A13 = a11A11 + a21A21 + a31A31
    where Aij is cofactor of aij.

Properties of Determinants

  • If rows are changed into columns and columns into rows, then the values of the determinant remain unaltered.
  • If any two-row (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.
  • If two rows (or two columns) in a determinant have corresponding elements that are equal, the value of the determinant is equal to zero.
  • If each element in a row (or column) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants.
  • If to each element of a line (row or column) of a determinant be added the equimultiples of the corresponding elements of one or more parallel lines, the determinant remains unaltered.
    i.e.Determinants - Introduction JEE Notes | EduRev
  • If each element in any row (or any column) of the determinant is zero, then the value of the determinant is equal to zero.
  • If a determinant D vanishes for x = a, then (x - a) is a factor of D, In other words, if two rows (or two columns) become identical for x = a, then (x-a) is a factor of D.

Determinants - Introduction JEE Notes | EduRev
Determinants - Introduction JEE Notes | EduRev

Area of triangle

Assume a triangle with vertices, (x1,y1), (x2,y2), (x3,y3) in a cartisean co-ordinate system as shown below.

Determinants - Introduction JEE Notes | EduRev

Then the area of the given triangle,Determinants - Introduction JEE Notes | EduRev 
It can be represented in the form of the determinant as:
Determinants - Introduction JEE Notes | EduRev

System of Linear Equation (Cramer's Rule)

Case 1: System of linear equations in two variables.

Let the system of equations be: 

a1x + b1 y + c1 = 0
a2x + b2 y + c2 = 0, where a1/a2 ≠ b1/b2

Solving by cross multiplication, we get,
Determinants - Introduction JEE Notes | EduRev
Determinants - Introduction JEE Notes | EduRev

Case 2: System of linear equations in three variables.

Let the system of equations be: 

a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3

Determinants - Introduction JEE Notes | EduRev
Simillarly for y = Δ2/Δ and z = Δ3

Determinants - Introduction JEE Notes | EduRev

Determinants - Introduction JEE Notes | EduRev

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