Matrix Algebra

Consistency

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 $b \neq 0$ . $\displaystyle$

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:

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

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

$\begin{displaymath} \left\{ \begin{array}{rrrrrrrr} & 3x_2 & -\,6x_3 & +\,6x... ... &+\,12x_3 & -\,9x_4 & +\,6x_5 & = & 15 \end{array} \right. \end{displaymath}$