Summary of Class 12 Maths Determinants

Summary of Class 12 Maths Determinants

 

Determinant

Corresponding to every square matrix A = [a_ij] of order n, we can associate a number called determinant of the square matrix A. It is denoted by det (A) or |A|.

Determinant of Matrix of Order 1

Let A = [a] be a matrix of order 1, then determinant of A is equal to a.

Determinant of Matrix of Order 2

Let A =  be a matrix of order 2, then |A| = a₁₁ a₂₂ – a₂₁ a₁₂  

Determinant of Matrix of Order 3

Let B =   be a matrix of order 3, then |B| = a₁₁(a₂₂ a₃₃ – a₃₂ a₂₃) – a₁₂(a₂₁

a₃₃ – a₃₁ a₂₃) + a₁₃(a₂₁ a₃₂ – a₃₁ a₂₂).

 

Properties of a Determinant

(i) The determinant of the product of matrices is equal to the product of their respective determinants, i.e. |AB| = |A| |B|, where A and B are a square matrix of the same order.

(ii) If A^T represents the transpose of a matrix A, then |A| = |A^T|

(iii) If A and B are square matrices of same order n, then |kAB| = kⁿ |A| |B|

 

Area of a Triangle

If (x_1, y₁), (x₂, y₂) and (x₃, y₃) be the vertices of a triangle, then its area is given by

                                                      △ =

 

Condition of Collinearity for Three Points

Three points A(x_1, y₁), B(x₂, y₂) and C(x₃, y₃) are collinear if and only if the area of triangle formed by these three points is zero.

That is,                                                     = 0

Minors

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.

Cofactors

Cofactor of an element a_ij, denoted by A_ij is defined by

A_ij = (−1)^{i + j} M_ij  , where M_ij is minor of a_ij.

 

Singular and Non-singular Matrix

For a square matrix A, if |A| = 0, then A is said to be a singular matrix and if |A| ≠ 0, then A is said to be a non-singular matrix.

 

Adjoint of a Matrix

The adjoint of a square matrix A is defined as the transpose of the matrix formed by cofactors of elements of A.

Let A = [a_ij]_{n × n} be a square matrix, then adjoint of A,

i.e.             adj (A) = C^T

where C = [a_ij] is the cofactor of matrix A.

The three steps involved in finding the adjoint of a square matrix A are:

·        Find the minors of all the elements of matrix A.

·        Find the cofactor matrix of A.

·        Find the adj (A) by taking the transpose of the cofactor matrix.

 

Properties of Adjoint of Square Matrix

6. |adj (adj A)| =

 

Inverse of a Matrix

Let A be a non-zero square matrix of order n and I be the identity matrix of order n.

Then, there exists a square matrix B of order n such that AB = BA = I. Such matrix B is called the inverse of matrix A. It is denoted by A^-1 and defined as

            A^-1 = adj (A) / IAI

 

Properties of Inverse of a Matrix

1. (A^-1) ^-1 = A

2. (A^-1) ^T = (A^T) ^-1

3. A A^-1 = A^-1 A = I

4. (AB)^-1 = B^-1A^-1

5. (kA)^-1 =   A^-1 , where k ≠ 0