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Q.
If the system of equations $2x + 3y = 8$, $7x - 5y + 3 = 0$, $4x - 6y + \lambda = 0$ has no solution, then $\lambda$
Determinants
Solution:
The given system of equations can be written as,
$2x + 3y + 0.Z=8 \quad\ldots\left(i\right)$
$7x - 5y + 0.z=-3\quad\ldots\left(ii\right)$
$4x - 6y + 0.z=-\lambda \quad\ldots\left(iii\right) $
The above equation can be written in matrix form as,
$\left[\begin{matrix}2&3&0\\ 7&-5&0\\ 4&-6&0\end{matrix}\right]\left[\begin{matrix}x\\ y\\ z\end{matrix}\right]=\left[\begin{matrix}8\\ -3\\ -\lambda\end{matrix}\right]$
or$\quad AX=B$, where
$A=\left[\begin{matrix}2&3&0\\ 7&-5&0\\ 4&-6&0\end{matrix}\right], X=\left[\begin{matrix}x\\ y\\ z\end{matrix}\right]$, and $B=\left[\begin{matrix}8\\ -3\\ -\lambda\end{matrix}\right]$
$\begin{vmatrix}2&3&0\end{vmatrix}$
The system has no solution if $\left(adj \,A \right)$ $B \ne O$
So, cofactor matrix of $A=\left[\begin{matrix}0&0&-22\\ 0&0&24\\ 0&0&-31\end{matrix}\right]$
$\therefore \quad\left(adj A\right) =\left[\begin{matrix}0&0&0\\ 0&0&0\\ -22&24&-31\end{matrix}\right]$
Now, (adj A )$B \ne O$
$\Rightarrow \quad\left[\begin{matrix}0&0&0\\ 0&0&0\\ -22&24&-31\end{matrix}\right]\left[\begin{matrix}8\\ -3\\ -\lambda\end{matrix}\right]\ne\left[\begin{matrix}0\\ 0\\ 0\end{matrix}\right]$
$\Rightarrow \quad\left[\begin{matrix}0+0+0\\ 0+0+0\\ -176-72+31\lambda\end{matrix}\right]\ne\left[\begin{matrix}0\\ 0\\ 0\end{matrix}\right]$
$\Rightarrow \quad-248+31\,\lambda \ne0 \Rightarrow 31\,\lambda \ne0 \,\Rightarrow \quad31\,\lambda \ne248\,\Rightarrow \,\lambda\ne8$