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Introduction

Determinants

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.

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

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

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

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.

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

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.

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

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

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

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

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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