Echelon Matrices

Linear systems that are in a certain special form are easy to solve. Have a look at an example of such a special linear system below:

The system is in an upper triangular form and can easily be solved by starting from the bottom (the last equation) and proceeding upwards (toward the first equation). This process is called back substitution. The second equation instantly gives us the value of $x_2=1$. Substituting the value $x_2$ back into the first equation, we get $x_1-2=1$. We get $x_1=3$, and the complete solution is $(x_1, x_2)=(3, 1)$. The augmented matrix of this triangular system of linear equations is considered an echelon matrix and is known as the echelon form. The generic definition of the echelon matrix with constraints is described below.

Echelon forms of a matrix

Echelon matrices are of two types:

• Row echelon
• Reduced row echelon

Every augmented matrix can be converted to either of the two echelon forms using a systematic algorithm. Such a matrix allows us to find possible solutions using back-substitution.

Row echelon form

A matrix is in the row echelon form ($ref$) if it meets the following requirements:

• The first non-zero number from the left, called the pivot, is always to the right of the pivot in the row above.
• Rows consisting of all zeros are at the bottom of the matrix.

A general $3\times4$ matrix in row echelon form is

$$\begin{bmatrix} \bold{a_{11}} & a_{12} & a_{13} & a_{14}\\ 0 & \bold{a_{22}} & a_{23} & a_{24}\\ 0 & 0 & 0 & \bold{a_{34}} \end{bmatrix}$$ ...