Mathematics - Determinant Question with Solution | TestHub

Given system of equations can be written in matrix form as $\mathrm{AX}=\mathrm{B}$ where $\mathrm{A}=\left(\begin{array}{lll}1& 2& 3\\ 1& 3& 5\\ 2& 5& a\end{array}\right)$ and $\mathrm{B}=\left(\begin{array}{l}6\\ 9\\ b\end{array}\right)$ Since, system is consistent and has infinitely many solutions $\therefore (\mathrm{adj}.\mathrm{A})\mathrm{B}=0$ $$\begin{array}{rl}& \Rightarrow \left(\begin{array}{ccc}3a-25& 15-2a& 1\\ 10-a& a-6& -2\\ -1& -1& 1\end{array}\right)\left(\begin{array}{l}6\\ 9\\ b\end{array}\right)=\left(\begin{array}{l}0\\ 0\\ 0\end{array}\right)\\ & \Rightarrow -6-9+b=0\Rightarrow b=15\\ & \text{ and }6(10-a)+9(a-6)-2(b)=0\\ & \Rightarrow 60-6a+9a-54-30=0\\ & \Rightarrow 3a=24\Rightarrow a=8\end{array}$$ Hence, $a=8,b=15$.