If the system of equations gathered ( -1)x+( -4)y+ z=5 x+( -1)y+( -4)z=7 ( +1)x+( +2)y-( +…

Question

If the system of equations

$\begin{gathered} (\lambda -1)x+(\lambda -4)y+\lambda z=5 \ \lambda x+(\lambda -1)y+(\lambda -4)z=7 \ (\lambda +1)x+(\lambda +2)y-(\lambda +2)z=9 \end{gathered}$

has infinitely many solutions, then ${\lambda }^{2}+\lambda $ is equal to

Accepted Answer

$\begin{gathered} \begin{gathered} \quad (\lambda -1)x+(\lambda -4)y+\lambda z=5 \ \lambda x+(\lambda -1)y+(\lambda -4)z=7 \ (\lambda +1)x+(\lambda +2)y-(\lambda +2)z=9 \ \quad D=|\begin{array}{ccc}\lambda -1 \quad \lambda -4 \quad \lambda \ \lambda \quad \lambda -1 \quad \lambda -4 \ \lambda +1 \quad \lambda +2 \quad -(\lambda +2)\end{array}|=0 \ (\lambda -3)(2\lambda +1)=0 \ {D}_{x}=|\begin{array}{ccc}5 \quad \lambda -4 \quad \lambda \ 7 \quad \lambda -1 \quad \lambda -4 \ 9 \quad \lambda +2 \quad -(\lambda +2)\end{array}|=0 \ 2(3-\lambda )(23-2\lambda )=0 \ \lambda =3 \ 	herefore {\lambda }^{2}+\lambda =9+3=12 \end{gathered} \end{gathered}$

Solutions · 1