Question

# The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is co

Vectors and spaces

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. $$\begin{bmatrix}1 & -2&6 \\0 & 0&0 \end{bmatrix}$$

2021-05-26

The augmented matrix does not contain a row in which the only nonzero entry appears in the last column. Therefore, this system of equations must be consistent.

Convert the augmented matrix into a system of equations.

$$x_1-2x_2=6$$
0=0

Solve for the leading entry for each individual equation. Determine the free variables, if any.
$$x_1=6+2x_2$$
$$x_2$$, free

Parameterize the free variables.
$$x_1=6+2t$$
$$x_2=t$$

Write the solution in vector form.
$$x=\begin{bmatrix}6 \\0 \end{bmatrix}+t\begin{bmatrix}2 \\1 \end{bmatrix}$$