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# Examine the consistency of the system of equations: $\quad x + 2y = 2$ $\quad 2x + 3y = 3$

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.
The given system of equation is

x+2y=2

2x+3y=3

This can be written in the form AX=B,

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

Hence $\begin{bmatrix}1 & 2\\2 & 3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\3\end{bmatrix}$

Now let us find the value of the determinant

|A|=3-2=1$\neq 0$

Hence A is non-singular.

Therefore $A^{-1}$ exists.

Hence the given system is consistent.

edited Feb 28, 2013
I dont personally like a solution in terms of different branch of Maths (=matrix algebra in this case). The preferred solution should be in regular algebra IMO. What if the student is not familiar with matrix algebra? Or what if his/her examiner expects the answer in regular algebra?  I understand they are all same, but the requirement for the student could be different. Only if a question can not be answered easily in the same branch of Math, we should choose other branches. (example: Derivation of Area of a circle). This is just my humble (non-mathematician's) opinion. I like you guys helping the knowledge seekers. Keep up the good work.