Mathematics - Determinant Question with Solution | TestHub

The given equations are
$\begin{array}{rl}& x+y+2z=6\\ & 2x+3y+az=a+1\\ & -x-3y+bz=2b,\text{ where }a,b,\in \mathbf{R}\end{array}$

For infinite many solutions:
$D=D_1=D_2=D_3=0$
$\begin{array}{rl}& \therefore D=\left|\begin{array}{ccc}1& 1& 2\\ 2& 3& a\\ -1& -3& b\end{array}\right|=2a+b-6\\ & D_1=\left|\begin{array}{ccc}6& 1& 2\\ a+1& 3& a\\ 2b& -3& b\end{array}\right|=12a+5b+ab-6\\ & D_2=\left|\begin{array}{ccc}1& 6& 2\\ 2& a+1& a\\ -1& 2b& b\end{array}\right|=-4a-3b-ab+2\\ & \text{ and }{D}_3=\left|\begin{array}{ccc}1& 1& 6\\ 2& 3& a+1\\ -1& -3& 2b\end{array}\right|=2a+2b-16\end{array}$
from above relations
$\begin{array}{rl}& a=-2,b=10\\ & \therefore 7a+3b=16\end{array}$