# Examine the consistency of the system of equations: $\quad x + 3y = 5$ $\quad 2x + 6y = 8$

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.
• (iii)If A is a singular matrix,then |A|=0.
• In this case we calculate (adj A)B.
• If (adj A)B$\neq 0$,then there is no solution and the system is said the system is said to be in consistent.
The given system can be written in the form AX=B,

$A=\begin{bmatrix}1 & 3\\2 & 6\end{bmatrix}\;X=\begin{bmatrix}x\\y\end{bmatrix}\;B=\begin{bmatrix}5\\8\end{bmatrix}$

Hence $\begin{bmatrix}1 & 3\\2 & 6\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\8\end{bmatrix}$

Now let us find the value of the determinant

|A|=6-6=0

Hence it is a singular matrix.

Now let us calculate (adj A)

(adj A) can be calculated by interchanging the elements $a_{11}$ and $a_{22}$ and then by changing symbols of $a_{12}$ and $a_{21}$

Therefore $(adj A)=\begin{bmatrix}6 & -3\\-2 & 1\end{bmatrix}$

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

Matrix multiplication can be obtained by multiplying the rows with the column.

Since $(adj A)B\neq 0$,the solution of the given system does not exist.hence it is consistent.

edited Feb 28, 2013