A matrix A is in row-echelon form if the first non-zero element of each row is 1 and each of these leading 1s is further right than the leading 1s of the preceding rows. Gaussian elimination will transform a matrix into row echelon form, and the row echelon form of the augmented matrix of a system of linear equations has the same set of solutions as the original, but in a form that is simple to solve.
The ref command transforms a matrix into a row echelon form of the matrix.
ref is typically used to solve a linear system of equations written in matrix form.
Example.
Solve the system:
| ⎧ ⎨ ⎩ |
|
Input:
Output:
| ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ |
Hence the solution is y=4 (from the last row) and x=−2 (substitute y in the first row).