If the system of equations [(-1)x+(-4)y+ z=5, x+(-1)y+(-4)z=7, (+1)x+(+2)y-(+2)z=9] has in…

Question

If the system of equations
$$(\lambda-1)x+(\lambda-4)y+\lambda z=5,\qquad \lambda x+(\lambda-1)y+(\lambda-4)z=7,\qquad (\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$$
has infinitely many solutions, then $\lambda$ is equal to:

Accepted Answer

$D=\begin{vmatrix}\lambda-1&\lambda-4&\lambda\\lambda&\lambda-1&\lambda-4\\lambda+1&\lambda+2&-(\lambda+2)\end{vmatrix}=0$ factors as $(\lambda-3)(2\lambda+1)=0$.

Checking $D_x=0$ (another required condition for infinitely many solutions) selects $\lambda=3$.

The quantity asked for (matching the source) is $\lambda^2+\lambda=9+3=12$.

Solutions · 1