Question

The sum of distinct values of $\lambda$ for which the system of equations

$$\begin{gathered} \left(\lambda -1\right)x+\left(3\lambda +1\right)y+2\lambda z=0 \\ \left(\lambda -1\right)x+\left(4\lambda -2\right)y+\left(\lambda +3\right)z=0 \\ 2x+\left(3\lambda +1\right)y+3\left(\lambda -1\right)z=0 \end{gathered}$$

has no solutions is

Answer 1

Find the sum of the distinct value of $\lambda$

The system of linear equation can be represented as $AX=B$

Where $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],A=\left[\begin{array}{ccc}\lambda -1& 3\lambda +1& 2\lambda \\ \lambda -1& 4\lambda -3& \lambda +3\\ 2& 3\lambda +1& 3\left(\lambda -1\right)\end{array}\right]$

$$\begin{gathered} \therefore \left|A\right|=\left|\begin{array}{ccc}\lambda -1& 3\lambda +1& 2\lambda \\ \lambda -1& 4\lambda -3& \lambda +3\\ 2& 3\lambda +1& 3\left(\lambda -1\right)\end{array}\right|=0 \\ {R}_2\to R_2-R_1 \\ {R}_3\to R_3-R_1 \\ \Rightarrow \left|\begin{array}{ccc}\lambda -1& 3\lambda +1& 2\lambda \\ 0& \lambda -3& -\lambda +3\\ 3-\lambda & 0& \left(\lambda -1\right)\end{array}\right|=0 \\ {C}_1\to C_1+C_3 \\ \Rightarrow \left|\begin{array}{ccc}3\lambda -1& 3\lambda +1& 2\lambda \\ -\lambda +3& \lambda -3& -\lambda +3\\ 0& 0& \left(\lambda -3\right)\end{array}\right|=0 \\ \Rightarrow \left(\lambda -3\right)\left[\left(3\lambda -1\right)\left(\lambda -3\right)-\left(3\lambda +1\right)\left(-\lambda +3\right)\right]=0 \\ \Rightarrow \left(\lambda -3\right)\left(3{\lambda }^2-10\lambda +3+3{\lambda }^2-8\lambda -3\right)=0 \\ \Rightarrow \left(\lambda -3\right)\left(6{\lambda }^2-18\lambda \right)=0 \\ \Rightarrow 6\lambda \left(\lambda -3\right)\left(\lambda -3\right)=0 \end{gathered}$$

Therefore $\lambda =0,3$

So the sum of the value of $\lambda =0+3=3$

Hence the sum of the distinct value of $\lambda$ for which the given system of linear equations has no solution is $3$.