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Mathematics Question on Determinants

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

Answer

The given system of equations is:
x + 3y = 5
2x + 6y = 8
The given system of equations can be written in the form of AX = B, where

A=[13\26]\begin{bmatrix}1&3\\\2&6\end{bmatrix},X=[x\y]\begin{bmatrix}x\\\y\end{bmatrix} and B=[5\8]\begin{bmatrix}5\\\8\end{bmatrix}

Now IAI=1(6)-3(2)=6-6=0
∴ A is a singular matrix.
Now (adjA)=[6−3−21]\begin{bmatrix}6&-3\\\\-2&1\end{bmatrix}

(adj B)=\begin{bmatrix}6&-3\\\\-2&1\end{bmatrix}$$\begin{bmatrix}5\\\8\end{bmatrix}=[30−24−10−8]\begin{bmatrix}30-24\\\\-10-8\end{bmatrix}=\begin{bmatrix}6\\\\-2\end{bmatrix}$$\neq 0

Thus, the solution of the given system of equations does not exist. Hence, the system of
equations is inconsistent.