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Examine the consistency of the system of equations:\[\] \[\quad x + 2y = 2 \] \[\quad 2x + 3y = 3\]

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  • (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
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.


answered Feb 26, 2013 by sreemathi.v
edited Feb 28, 2013 by vijayalakshmi_ramakrishnans
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.

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