How is Gaussian elimination performed?

Gaussian elimination is a systematic method used to solve linear equations, find the rank of a matrix, and calculate the inverse of an invertible matrix. The goal of Gaussian elimination is to transform a given matrix into row-echelon form (REF) or reduced row-echelon form (RREF) through a sequence of operations.

The process involves applying three fundamental types of row operations:

1. Swapping two rows.
2. Multiplying a row by a non-zero scalar.
3. Adding or subtracting multiples of one row to another row.

The correct answer highlights that Gaussian elimination is specifically achieved by applying these row operations to the given matrix to systematically eliminate variables, ultimately leading to a clearer strategy for solution or further analysis. Achieving row-echelon form means that each leading coefficient (the first non-zero number from the left in each row) is positioned further to the right than the leading coefficient of the previous row, and all entries below a leading coefficient are zero.

While adding rows can be part of the process, it doesn't encapsulate the entire method as comprehensively as the correct answer does. Similarly, multiplying columns or reorganizing the matrix into standard form does not accurately represent the essence of Gaussian elimination, which focuses specifically on manipulating rows to facilitate solving