# Examine the consistency of the system of equations: $\quad x + y + z = 1$ $\quad 2x + 3y +2z = 2$ $\quad ax + ay +2az = 4$

Toolbox:
• (i)A matrix is said to be invertible if inverse exists.
• (ii)If A is a non singular matrix such that
• AX=B
• then X=$A^{-1}B$
• using this we can solve system of equations,which has unique solutions.
• (iii)If A is a singular matrix,then |A|=0.
• If (adj A)B$\neq 0$,then there is no solution and the system is said the system is said to be in consistent.
This can be written in the form AX=B

$\begin{bmatrix}1 & 1& 1\\2 & 3& 2\\a & a &2a\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\2\\4\end{bmatrix}$

Where $A=\begin{bmatrix}1 & 1& 1\\2 & 3& 2\\a & a&2a\end{bmatrix}\;X=\begin{bmatrix}x\\y\\z\end{bmatrix}\;B=\begin{bmatrix}1\\2\\4\end{bmatrix}$

Let us evaluate the value of determinant |A| by expanding along $R_1$

$|A|=1(3\times 2a-2\times a)-1(2\times 2a-2\times a)+1(2\times a-3a)$

$\;\;\;=6a-2a-4a+2a+2a-3a$

$\;\;\;=a\neq 0$

Therefore A is non singular.

Hence $A^{-1}$ exists.

Hence the given system of equation is consistent.

edited Feb 28, 2013