determinants class 12 notes

Determinants for Class 12 Notes

Definition: Every square matrix A of the order n, can associate a number called determinants of the square matrix A.

Determinant of the order one (1×1)

Consider a matrix A = [a], then the determinant of the matrix is equal to a.

Determinant of the order Two (2×2)

If the order of the matrix is 2, then the determinants is defined matrix A, where A is-

Similarly we can find the determinant of the order 3×3

Determinant of the Order Three (3×3)

Suppose a matrix A of order three is given:

Then the determinant for a 3×3 matrix is given by;

|A| = a₁₁ a₂₂ a₃₃ – a₁₁ a₂₃ a₃₂ – a₁₂ a₂₁ a₃₃ + a₁₂ a₂₃ a₃₁ + a₁₃ a₂₁ a₃₂ – a₁₃ a₃₁ a₂₂

Properties of Determinants

Let us now look on to the properties of the Determinants which is discussed in determinants for class 12:

Property 1- The value of the determinant remains unchanged if the rows and columns of a determinant are interchanged.

Property 2- If any two rows (or columns) of determinants are interchanged, then sign of determinants changes.

Property 3- If any two rows or columns of a determinant are equal or identical, then the value of the determinant is 0.

Property 4- If each element of a row or a column is multiplied by a constant value k, then the value of the determinant originally obtained is multiplied with k.

Area of Triangle Using Determinant

We have already learned, that the area of a triangle whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃) is given by;

A = 1/2[x₁(y₂–y₃) + x₂(y₃–y₁) + x₃(y₁–y₂)]

Now, we can write the above expression in the form of the determinant as;

Minors and Cofactors

Minor of an element a_ij of a determinant is the determinant obtained by deleting its ‘ith’ row and ‘jth’ column in which element a_ij lies. Minor of an element a_ij is denoted by M_ij. Minor of an element of a  determinant of order n(n ≥ 2) is a determinant of order n – 1. Or in simple words we can say, the minor of any element in a determinant is the determinant of just the single element. Suppose,

is a determinant and we have to find the minor of element ‘f’, then, we can write;

M_f = a.h – b.g

Cofactor of an element a_ij in a determinant is defined by;

A_ij= (-1)^i+jM_ij

Apart from these topics, there are few more topics covered in chapter 4 of class 12 Maths, such as;

• adjoint and inverse of a square matrix
• consistency and inconsistency of linear equations
• the solution of linear equations in two or three variables using the inverse of a matrix