Determinants-System of Linear Equations

Relationship Between Determinants and Systems of Equations:

1. Matrix Representation:

Consider a system of linear equations:

a11x1+a12x2++a1nxn=b1

a21x1+a22x2++a2nxn=b2

an1x1+an2x2++annxn=bn

This system can be represented in matrix form as AX=B where:

  • A is the coefficient matrix containing coefficients aij.
  • X is the column matrix of variables xi.
  • B is the column matrix of constants bi.

2. Determinants and Solutions:

  • Cramer's Rule: For a system of equations represented by AX=B, the solution for xi can be expressed in terms of determinants of matrices derived from A by replacing the ith column with matrix B.

  • Non-Zero Determinant: If the determinant of matrix A(A) is non-zero, the system has a unique solution. Each variable xi can be determined uniquely.

  • Zero Determinant: If A=0:

    • The system may have infinitely many solutions or no solutions at all.
    • The columns of matrix A are linearly dependent, indicating redundancy or inconsistency in the equations.

Example:

Let's consider a system of linear equations:

2x+3y=8

4x2y=2

Representing this system in matrix form:

A=[2342]

 X=[xy]

 B=[82]

  • Calculate the determinant of matrix A (A).
  • Determine if the system has a unique solution based on the value of A.
  • Apply Cramer's Rule to find the solution if A is non-zero.

Cramer's Rule:

Given a system of linear equations:

a11x1+a12x2++a1nxn=b1

a21x1+a22x2++a2nxn=b2

an1x1+an2x2++annxn=bn

Represented in matrix form: AX=B

Where:

  • A is the coefficient matrix containing coefficients aij.
  • X is the column matrix of variables xi.
  • B is the column matrix of constants bi.

Using Cramer's Rule:

  1. Calculating Determinants:

    • Compute the determinant of matrix A denoted as A.
    • For each variable xi, replace the ith column of matrix A with matrix B to create a new matrix, say Ai.
    • Calculate the determinant of matrix Ai denoted as Ai.
  2. Solving for Variables:

    • The solution for variable xi is given by: xi=AiA
  3. Unique Solution:

    • If A0, Cramer's Rule provides a unique solution for each variable xi.
    • If A=0 or Ai=0 for some i, Cramer's Rule cannot be applied or the system may have no unique solution.

Example:

Let's solve the system of linear equations using Cramer's Rule:

3x+2y=11

2xy=3

Representing this system in matrix form:

A=[3221] X=[xy] B=[113]

  1. Calculate the determinant of matrix A (A).
  2. Calculate determinants A1 and A2.
  3. Use Cramer's Rule to find x and y.