What is the goal of performing row operations in Gaussian elimination?

The goal of performing row operations in Gaussian elimination is to simplify the matrix into reduced row-echelon form. This process involves applying three types of row operations: swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row to another. By doing so, the elimination process systematically zeros out coefficients below the leading ones, ultimately transforming the matrix into a simplified form that allows for easier back substitution to find solutions to the associated linear system.

Achieving reduced row-echelon form ensures that each leading entry in a row is 1, and that these leading 1s are the only non-zero entries in their respective columns. This clarity provides a straightforward interpretation of the solutions to the system, whether they be unique, infinitely many, or none.

Other approaches, such as creating a diagonal matrix or calculating the determinant, are not the primary objectives of Gaussian elimination. Although multiplying rows by a constant is one of the operations involved in the process, it is only a part of the larger goal of achieving the reduced row-echelon form. Thus, the correct response highlights the overarching aim of the method.