﻿ Class 12 NCERT Math Solution
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Introduction

Introduction of Determinant

To every square matrix A = [aᵢⱼ] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where aᵢⱼ = (i, j)th element of A. It is also denoted by |A| or det A or Δ. Example Note : (i) For matrix A, |A| is read as determinant of A and not modulus of A.
(ii) Only square matrices have determinants.

Determinant of a matrix of order one

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

Determinant of a matrix of order two Determinant of a matrix of order three  Property-1 : The value of the determinant remains unchanged if its rows and columns are interchanged. Property-2 : If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes. Property-3 : If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then value of determinant is zero.

If we interchange the identical rows (or columns) of the determinant Δ, then Δ does not change. However, by Property 2, it follows that Δ has changed its sign Therefore Δ = – Δ or Δ =0

Property-4 : If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k. Property-5 : If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants. Property-6 : If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation Rᵢ → Rᵢ + kRⱼ or Cᵢ → Cᵢ + kCⱼ. Area of a Triangle

The area of a triangle whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃), is given by the expression 1/2[x₁(y₂–y₃) + x₂ (y₃–y₁) + x₃ (y₁–y₂)]. Now this expression can be written in the form of a determinant as Minors

Minor of an element aᵢⱼ of a determinant is the determinant obtained by deleting its ith row and jth column in which element aᵢⱼ lies. Minor of an element aᵢⱼ is denoted by Mᵢⱼ.

Cofactors

Cofactor of an element aᵢⱼ, denoted by Aᵢⱼ is defined by Aᵢⱼ = (–1)ⁱ⁺ᴶ Mᵢⱼ, where Mᵢⱼ is minor of aᵢⱼ.

Adjoint and Inverse of a Matrix

The adjoint of a square matrix A = [aᵢⱼ]n × n is defined as the transpose of the matrix [Aᵢⱼ]n × n, where Aᵢⱼ is the cofactor of the element aᵢⱼ. Adjoint of the matrix A is denoted by adj A.

Singular matrix

A square matrix A is said to be singular if A = 0.

Non-Singular matrix

A square matrix A is said to be non-singular if A ≠ 0

Consistent system

A system of equations is said to be consistent if its solution (one or more) exists.

Inconsistent system

A system of equations is said to be inconsistent if its solution does not exist.

Solution of system of linear equations using inverse of a matrix

Let us express the system of linear equations as matrix equations and solve them using inverse of the coefficient matrix. Case I : If A is a nonsingular matrix, then its inverse exists. Now This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method.
Case II : If A is a singular matrix, then |A| = 0. In this case, we calculate (adj A) B. If (adj A) B ≠ O, (O being zero matrix), then solution does not exist and the system of equations is called inconsistent. If (adj A) B = O, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.

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