**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) $\rho(A) = \rho[A,B] = n$ where $n$ is the number of unknowns $\Rightarrow$ consistency with unique solution. (c) If $\rho(A)=\rho[A,B] < n,$ 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 & 1 & 1 \\ 1 & 2 & 3 \\ 0 & 1 & 2 \end{bmatrix} \begin{bmatrix}x \\ y \\ z \end{bmatrix} = \begin{bmatrix}7 \\ 18 \\ 6 \end{bmatrix}$

$AX=B$

The angmented matrix $[A,B] = \begin{bmatrix}1 & 1 & 1 & 7 \\ 1 & 2 & 3 & 18 \\ 0 & 1 & 2 & 6 \end{bmatrix}$

$ \sim \begin{bmatrix}1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 11 \\ 0 & 1 & 2 & 6 \end{bmatrix}R_2 \rightarrow R_2-R_1$

$ \sim \begin{bmatrix}1 & 1 & 1 & 7 \\ 0 & 1 & 2 & 11 \\ 0 & 0 & 0 & -5 \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.