### 15.7.2 Gauss reduction of a matrix

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 takes A, a matrix.
- ref(A) returns a row echelon form of a matrix.

ref is typically used to solve a linear system of equations written in matrix form.

#### Example

Solve the system:

| ⎧ ⎨ ⎩ |
3x+y | = | −2
3x +2y | = | 2

ref([[3,1,-2],[3,2,2]])

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ |
1 |
1
3
− |
2
3
0 | 1 | 4
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Hence the solution is y=4 (from the last row) and x=−2 (substitute y in the first row).