# Examine the consistency of the following system of equation. If it is consistent than solve the same. $x-4y+7z=14\;,3x+8y-2z=13\;,7x-8y+26z=5$

Toolbox:
• Rank method for finding consistency of a system of $m$ equation in $n$ unknowns : (i) Write the matrix equation $AX = B$ (ii) Find the angmented matrix $[ A, B]$ (iii) Find the ranks of A and [A,B] by elementary row transformations. (iv) (a) $\rho(A) \neq \rho[A,B] \Rightarrow$ inconsistency with no solution (b) $\rhoy(A) = \rho[A,B] = n$ where $n$ is the number of unknowns $\Rightarrow$ consistency with unique solution. (c) If $\rhop(A)=\rho[A,B] < n,$ the system is consistent with infinite number of solutions. (d) A system of homogeneous linear equations is always consistent, if $\rho(A) = n$ then the only solution is the trivial solution. If $\rho(A) < n$ then the system has nontrivial solutions in addition to the trivial solutions.
Step 1
The matrix equation is $\begin{bmatrix} 1 & -4 & 7 \\ 3 & 8 & -2 \\ 7 & -8 & 26 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 14 \\ 13 \\ 5 \end{bmatrix}$
$AX = B$
The angmented matrix
$[A,B] = \begin{bmatrix} 1 & -4 & 7 & 14 \\ 3 & 8 & -2 & 13 \\ 7 & -8 & 26 & 5 \end{bmatrix}$
$\sim \begin{bmatrix} 1 & -4 & 7 & 14 \\ 0 & 20 & -23 & -29 \\ 0 & 20 & -23 & -93 \end{bmatrix}R_2 \rightarrow R_2-3R_1$
$R_3 \rightarrow R_3-7R_1$
$\sim \begin{bmatrix} 1 & -4 & 7 & 14 \\ 0 & 20 & -23 & -29 \\ 0 & 0 & 0 & -64 \end{bmatrix} R_3 \rightarrow R_3-R_2$
Step 2
From the last equivalent matrix, which is in echelon form, it can be seen that $\rho(A)=2$ (2 nonzero rows) and $\rho[A,B]=3$ (3 nonzero rows).
$\rho(A) \neq \rho[A,B]. \therefore$ the system is inconsistent with no solution.

edited Jun 3, 2013