Question

Examine the consistency of the system of equations$x+3y=5$, $2x+6y=8$

Answer 1

Given system of equations

$x+3y=5$
$2x+6y=8$

This can be written as

$AX=B$

where $A=\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right],B=\left[\begin{array}{c}5\\ 8\end{array}\right]$

Here, $|A|=6-6=0$

Since, $|A|=0$

Hence, the system has either infinitely many solution (consistent) or no solution (inconsistent)

Now, we will find $\left(adjA\right)B$

$C_{11}=(-1{)}^{1+1}6=6$

$C_{12}=(-1{)}^{1+2}2=-2$

$C_{21}=(-1{)}^{2+1}3=-3$

$C_{22}=(-1{)}^{2+2}1=1$

So, the co-factor matrix is $C=\left[\begin{array}{cc}6& -2\\ -3& 1\end{array}\right]$

$\Rightarrow adjA=C^T=\left[\begin{array}{cc}6& -3\\ -2& 1\end{array}\right]$

Now, $\left(adjA\right)B=\left[\begin{array}{cc}6& -3\\ -2& 1\end{array}\right]\left[\begin{array}{c}5\\ 8\end{array}\right]$

$=\left[\begin{array}{c}30-24\\ -10+8\end{array}\right]$

$=\left[\begin{array}{c}6\\ -2\end{array}\right]$

$\Rightarrow \left(adjA\right)B\ne O$

Hence, the system of equation is inconsistent.