Exe. 3.5 Complete | Homogeneous Linear Equations | Cramer’s Rule | Punjab Text Book
Welcome to our YouTube video on the topic "System of Linear Equations, Homogeneous Linear Equations, Non-Trivial Solution, Non-Homogeneous Linear Equations, and Cramer's Rule." In this video, we will walk you through a comprehensive explanation of these fundamental concepts in linear algebra, specifically focusing on Exercise 3.5 from the Punjab Text Book Board. We will start by introducing the concept of a system of linear equations and its significance in various fields, such as engineering, physics, and economics. You will learn how to represent a system of linear equations using matrix notation and understand the concept of solutions, including unique, infinite, and inconsistent solutions. Next, we will delve into homogeneous linear equations and discuss their characteristics. You will understand the concept of a non-trivial solution and learn how to determine whether a homogeneous system has a non-trivial solution or only the trivial solution. Moving forward, we will explore non-homogeneous linear equations and their solutions. You will discover the difference between homogeneous and non-homogeneous systems and learn how to find the general solution of a non-homogeneous system using the method of Gaussian elimination. Furthermore, we will introduce you to Cramer's Rule, an alternative method for solving systems of linear equations. You will understand the conditions under which Cramer's Rule can be applied and learn the step-by-step process to use determinants to find the solutions. Throughout the video, we will provide clear explanations, example problems, and practical tips to enhance your understanding of these topics. By the end of this video, you will have a solid grasp of system of linear equations, homogeneous and non-homogeneous systems, non-trivial solutions, and Cramer's Rule, empowering you to tackle Exercise 3.5 from the Punjab Text Book Board with confidence. Here's an example problem that illustrates the concept of non-trivial solutions in homogeneous linear equations: Consider the following system of linear equations: 2x - 3y + z = 0 4x - 6y + 2z = 0 x - 2y + z = 0 To determine whether this system has non-trivial solutions, we can write it in matrix form as: AX = 0 where A is the coefficient matrix and X is the column vector of variables (x, y, z). The coefficient matrix A is: | 2 -3 1 | | 4 -6 2 | | 1 -2 1 | To find the non-trivial solutions, we need to solve the homogeneous system by reducing the matrix A to its row-echelon form or performing Gaussian elimination. Using Gaussian elimination, we can perform the following row operations to simplify the matrix A: Divide Row 1 by 2: (1/2)R1 -- R1 Multiply Row 1 by 4, and subtract it from Row 2: -4R1 + R2 -- R2 Multiply Row 1 by 1, and subtract it from Row 3: -R1 + R3 -- R3 After performing these row operations, the matrix A becomes: | 1 -3/2 1/2 | | 0 0 0 | | 0 0 0 | Now, we can see that the second and third rows of the matrix are identical and consist of all zeros. This indicates that we have an infinite number of solutions. To express the solutions, we can parameterize the variables: x = t y = s z = t where t and s are arbitrary constants. Therefore, the system has non-trivial solutions, represented by an infinite number of solutions parametrized by t and s. This example illustrates that non-trivial solutions in homogeneous linear equations occur when the system of equations has an infinite number of solutions, indicating that there is more than just the trivial solution (where all variables are equal to zero). All will be done but with the passage of time. #logicalzahidmahmood #maths #matrix
