Consistency

A system of linear equations is consistent (that is, it has a solution, or general solutions) if and only if an echelon matrix has no row of the form

$\displaystyle [\:0 \:\cdots\cdots\: 0\:\: b\:]$    with $ b \neq 0$.

Thus, you should check for consistency as soon as an echelon form is obtained. Then proceed the row reduction to produce a REF only when it is consistent.

EXAMPLE 4. Determine if the following system is consistent:

\begin{displaymath}
\begin{array}{rrrr}
& x_2 & -  4x_3 & = 8 \\
2x_1 & -  ...
... + 2x_3 & = 1 \\
5x_1 & -  8x_2 & + 7x_3 & = 1
\end{array}\end{displaymath}

EXAMPLE 5, Determine the existence and uniqueness of the solutions to the following system:

\begin{displaymath}
\begin{array}{rrrrrrrr}
& 3x_2 & - 6x_3 & + 6x_4 & + 4x_...
...1 & -  9x_2 &+ 12x_3 & - 9x_4 & + 6x_5 & = & 15
\end{array}\end{displaymath}



Department of Mathematics
Last modified: 2005-10-21