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It involves transforming a system of equations into an equivalent system, called a row-echelon form, which can easily be solved. This method relies on arithmetic operations such as row transformations, including multiplication and adding multiple rows together. The process starts by writing the system of equations as an augmented matrix, where the coefficients of the variables and the constant terms are organized in a rectangular arrangement. Then, a series of operations are performed to eliminate terms below and above certain pivot positions in the matrix, resulting in a triangular matrix. The values of the variables can then be easily obtained by back substitution. Gaussian elimination has numerous applications in various fields, including physics, engineering, and computer science. It is commonly used for solving systems of linear equations, determining the rank of a matrix, finding the inverse of a matrix, and solving problems related to linear transformations. The method is considered efficient and reliable when dealing with large systems of equations. Despite its advantages, Gaussian elimination does have limitations. It may encounter numerical stability problems when working with ill-conditioned matrices, leading to inaccuracies in the solutions. Additionally, it may not work for systems of equations that have no solutions or infinite solutions. The history of Gaussian elimination dates back to ancient China and the third century BCE, where similar methods were used to solve systems of linear equations. However, the method as it is known today was developed by the German mathematician Carl Friedrich Gauss in the early 19th century. Since then, the technique has been refined and improved upon by many mathematicians, leading to various algorithms and variations of Gaussian elimination.