# Gauss reduction of a matrix: ref

### 5.56.2 Gauss reduction of a matrix: ref

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 one argument:
  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

Input:

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

Output:

⎡
⎢
⎢
⎢
⎢
⎣

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).