Examine the consistency of the system of equations: $\quad 2x -y = 5$ $\quad x + y = 4$

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.
The given system of equation is

2x-y=5

x+y=4

This can be written in the form AX=B,

$A=\begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}\;X=\begin{bmatrix}x\\y\end{bmatrix}\;B=\begin{bmatrix}5\\4\end{bmatrix}$

Hence $\begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\4\end{bmatrix}$

Now let us find the value of the determinant

|A|=2-(-1)=2+1=3$\neq 0$

Hence A is non-singular.

Therefore $A^{-1}$ exists.

Hence the given system is consistent.

edited Feb 27, 2013