Example of solving three linear systems \(Ax=b\) (nonhomogeneous systems) with the same coefficients.

This Linear Algebra video tutorial provides a basic introduction into the Gauss-Jordan elimination which is a process that involves elementary row operations with \(3 \times 3\) matrices which allows you to solve a system of linear equations with \(3\) variables \((x, y, z)\).

❖ Solve a linear system \(Ax=b_1\) and \(Ax=b_2\) by using a Reduced Row Echelon Form (RREF). (Sometimes, they called this method as Gauss-Jordan elimination ( or Gauss-Jordan reduction) method).

❖ To solve a linear system of equations by Gauss Jordan elimination, we have to put it in RREF.

So, you need to convert the system of linear equations into an augmented matrix \([ A | b_1 | b_2 | b_3 ]\) and use matrix row operations to convert the \(3 \times 3\) matrix into the RREF. You can easily determine the answers once you convert to the RREF.

❖ We have solved the two systems (\(Ax=b_1, Ax=b_2,\) and \(Ax=b_3\)) in the following way:

$$[ A | b_1 | b_2 | b_3 ] to [\text{REFF} | c_1 | c_2 | c_3 ] $$

(\(b_1, b_2\) and \(b_3\) vectors) changed to (\(c_1, c_2\) and \(c_3\) vector) because we have done RREF for the augmented matrix \([ A | b_1 | b_2 | b_3 ]\).