Problem 55 - College Algebra

Question

What is the difference between Gaussian elimination and Gauss-Jordan elimination?

Step-by-Step Solution

Answer

Both Gaussian and Gauss-Jordan elimination are methods used to solve systems of linear equations. The main difference lies in the format of the resulting matrix: Gaussian elimination results in an echelon form while Gauss-Jordan results in a reduced echelon form, eliminating the need for manual back substitution to solve for the variables.

Step 1: Explain Gaussian Elimination

Gaussian elimination is a method for solving matrix equations of the form \(Ax=B\). The main idea of the method is to perform elementary operations on the rows of the matrix \(A\) until it is in row-echelon form. Once the row-echelon form of the matrix is obtained, the system of equations can be solved sequentially - generally starting from the last one and moving up.

Step 2: Explain Gauss-Jordan Elimination

Gauss-Jordan elimination is an extension of Gaussian elimination. Like Gaussian elimination, Gauss-Jordan elimination aims to simplify a matrix to solve the system of equations. However, the end goal of Gauss-Jordan elimination is to make the matrix into reduced row-echelon form (not just row-echelon form). This is achieved by not only making the elements below the diagonal to be zero (as in Gaussian elimination) but also making the elements above the diagonal to be zero. This way, the augmented matrix results in an identity matrix.

Step 3: Difference between Gaussian and Gauss-Jordan Elimination

The major difference between Gaussian and Gauss-Jordan elimination lies in the format of the end result: Gaussian elimination results in a row-echelon form of the matrix, and requires back substitution to get the solution; Gauss-Jordan elimination continues the process to achieve a reduced row-echelon form, which directly gives the solutions for all variables in the linear equations.