Matrix and Determinant - System of linear equation and Rank of a matrix

  • Introduction to Systems of Linear Equations
    • Definition of a system of linear equations
    • Examples of systems of linear equations
  • Solving Systems of Linear Equations by Elimination Method
    • Steps to solve a system of linear equations using elimination method
    • Example: Solve the system of equations using elimination method:
      • 2x + 3y = 7
      • 4x - 5y = 6
  • Solving Systems of Linear Equations by Substitution Method
    • Steps to solve a system of linear equations using substitution method
    • Example: Solve the system of equations using substitution method:
      • 3x + 2y = 8
      • 2x - 4y = -10
  • Solving Systems of Linear Equations by Matrix Method
    • Introduction to matrix representation of systems of linear equations
    • Example: Solve the system of equations using matrix method:
      • x + 2y - z = 1
      • 2x - y + z = -2
      • 3x + 4y - 3z = 3
  • Rank of a Matrix
    • Definition of rank of a matrix
    • Method to find the rank of a matrix
  • Properties of Matrix Rank
    • Important properties related to matrix rank
    • Examples illustrating the properties of matrix rank
  • Solving Systems of Linear Equations using Rank of a Matrix
    • Introduction to solving systems of linear equations using rank of a matrix
    • Example: Solve the system of equations using rank of a matrix:
      • 2x + 3y - z = 6
      • x - 2y + 3z = 1
      • 3x + 9y - 12z = 9
  • Homogeneous Systems of Linear Equations
    • Definition of homogeneous systems of linear equations
    • Conditions for homogeneous systems of linear equations to have non-trivial solutions
  • Solving Homogeneous Systems of Linear Equations
    • Steps to solve homogeneous systems of linear equations
    • Example: Solve the homogeneous systems of equations:
      • x + 2y - 3z = 0
      • 2x + 4y - 6z = 0
      • 3x + 6y - 9z = 0
  • Summary and Key Points
    • Recap of important concepts covered in this lecture
    • Key points to remember about systems of linear equations and rank of a matrix

Rank of a Matrix

  • The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
  • It is denoted by the symbol “r”.
  • The rank of a matrix can be determined by performing Gaussian elimination or by using the determinant method.
  • The rank of a matrix gives important information about the system of linear equations it represents.

Properties of Matrix Rank

  • The rank of a matrix cannot exceed the number of rows or columns in the matrix.
  • If the rank of a matrix is equal to the number of rows or columns, the matrix is said to be of full rank.
  • If the rank of a matrix is less than the number of rows or columns, the matrix is said to be rank deficient.
  • If a matrix is square and of full rank, it is invertible.
  • The rank of a matrix is equal to the maximum number of linearly independent rows or columns.

Solving Systems of Linear Equations using Rank of a Matrix

  • The rank of a coefficient matrix can be used to determine the number of solutions for a system of linear equations.
  • If the rank of the coefficient matrix is equal to the rank of the augmented matrix, then the system has a unique solution.
  • If the rank of the coefficient matrix is less than the rank of the augmented matrix, then the system has infinitely many solutions.
  • If the rank of the coefficient matrix is less than the number of variables, then the system has no solution.

Example: Solving a System of Equations using Rank of a Matrix

Consider the system of equations: ``

2x + 3y - z = 6 x - 2y + 3z = 1

3x + 9y - 12z = 9 We can represent this system in matrix form as Ax = b, where A = | 2 3 -1 | | 1 -2 3 | | 3 9 -12 | x = | x | | y | | z | b = | 6 | | 1 | | 9 | ``

Example: Solving a System of Equations using Rank of a Matrix (contd.)

By performing Gaussian elimination, we can find the rank of the coefficient matrix A. Using elementary row operations, we can reduce A to its row echelon form: | 2 3 -1 | | 2 3 -1 | | 1 -2 3 | => | 0 -7 4 | | 3 9 -12 | | 0 0 0 | The rank of the coefficient matrix A is 2, and the rank of the augmented matrix (A|b) is also 2.

Example: Solving a System of Equations using Rank of a Matrix (contd.)

Since the rank of the coefficient matrix is equal to the rank of the augmented matrix, the system has a unique solution. By back substitution, we can find the values of x, y, and z: ``

2x + 3y - z = 6 x = 1

0x - 7y + 4z = 0 => y = 2

0x + 0y + 0z = 0 z = 0 `` Therefore, the system of equations has a unique solution: x = 1, y = 2, z = 0.

Homogeneous Systems of Linear Equations

  • A homogeneous system of linear equations is a system in which the right-hand side of each equation is zero.
  • In other words, a homogeneous system can be written as Ax = 0, where A is the coefficient matrix and x is the vector of variables.
  • Every homogeneous system has at least one solution, known as the trivial solution, where all variables are equal to zero.
  • Homogeneous systems can also have non-trivial solutions, where at least one variable is non-zero.

Solving Homogeneous Systems of Linear Equations

  • To solve a homogeneous system of linear equations, we need to find the non-trivial solutions.
  • This can be done by setting up the augmented matrix (A|0) and performing Gaussian elimination to find the row echelon form.
  • The non-zero variables in the row echelon form will be the parameters in the non-trivial solutions.
  • By expressing the remaining variables in terms of the parameters, we can obtain the general solution to the homogeneous system.

Example: Solving a Homogeneous System of Equations

Consider the homogeneous system of equations: `` x + 2y - 3z = 0

2x + 4y - 6z = 0

3x + 6y - 9z = 0 We can represent this system in matrix form as Ax = 0, where A = | 1 2 -3 | | 2 4 -6 | | 3 6 -9 | x = | x | | y | | z | ``

Example: Solving a Homogeneous System of Equations (contd.)

By performing Gaussian elimination and reducing the coefficient matrix A to its row echelon form, we get: | 1 2 -3 | | 1 2 -3 | | 0 0 0 | => | 0 0 0 | | 0 0 0 | | 0 0 0 | In the row echelon form, we can see that the first two variables (x and y) are the parameters, while the third variable (z) is a free variable. Therefore, the general solution to the homogeneous system of equations is: x = -2y + 3z y = y z = z

Example: Solving a Homogeneous System of Equations (contd.)

Using the general solution obtained previously, we can find specific solutions for the variables. For example, let’s set the parameter y = 1 and z = 2: x = -2(1) + 3(2) = 4 y = 1 z = 2 Therefore, one possible non-trivial solution to the homogeneous system of equations is: x = 4, y = 1, z = 2

Summary and Key Points

  • Systems of linear equations can be solved using elimination, substitution, or matrix methods.
  • The rank of a matrix represents the maximum number of linearly independent rows or columns.
  • The rank of a matrix is useful in determining the number of solutions to a system of linear equations.
  • Homogeneous systems of linear equations always have the trivial solution, and can also have non-trivial solutions.
  • Non-trivial solutions of homogeneous systems can be found by setting up and solving the augmented matrix.

Key Points (contd.)

  • The determinant method can be used to find the rank of a matrix.
  • The number of solutions to a system of linear equations depends on the rank of the coefficient matrix and the augmented matrix.
  • If the rank of the coefficient matrix is equal to the rank of the augmented matrix, the system has a unique solution.
  • If the rank of the coefficient matrix is less than the rank of the augmented matrix, the system has infinitely many solutions.
  • If the rank of the coefficient matrix is less than the number of variables, the system has no solution.

Final Thoughts

  • Understanding systems of linear equations and the rank of a matrix is essential in various fields such as engineering, economics, and computer science.
  • These concepts provide a powerful tool to solve real-world problems and analyze mathematical models.
  • Practice solving different types of systems of linear equations to enhance your understanding and problem-solving skills.
  • Don’t forget to revise the properties and methods discussed in this lecture before your exams.

Practice Problems

  1. Solve the system of linear equations:
    • 3x + 2y + z = 7
    • 2x - y + 3z = 10
    • x + 3y - 2z = 6
  1. Find the rank of the matrix: A = | 1 2 3 | | 2 4 6 | | 3 6 9 |
  1. Solve the homogeneous system of equations:
    • 2x + 3y - z = 0
    • x - 4y + 3z = 0
    • 3x + 5y - 6z = 0

Practice Problems (contd.)

  1. Determine the number of solutions for the system of equations: 5x + 2y = 10 2x + y = 5
  1. Find the general solution for the homogeneous system of equations: x + 3y - z = 0 2x - y + 2z = 0 4x + y - 2z = 0
  1. Solve the system of linear equations using matrix method: 2x + y + z = 2 x - y + z = 0 -x + 2y + 3z = 5

Further Reading

  • “Linear Algebra and Its Applications” by David C. Lay, Steven R. Lay, and Judi J. McDonald.
  • “Matrix Methods: Applied Linear Algebra” by Richard Bronson.
  • “Introduction to Linear Algebra” by Gilbert Strang.
  • Online resources and video tutorials on matrix and determinant topics.

Conclusion

In this lecture, we covered the topics of systems of linear equations and the rank of a matrix. We discussed various methods for solving systems of equations, including elimination, substitution, and matrix methods. We also explored the concept of matrix rank and its importance in determining the number of solutions to systems of equations. Lastly, we learned about homogeneous systems and how to find their non-trivial solutions using the rank of a matrix. It is important to practice these concepts through problem-solving to strengthen your understanding.