Let the system of linear equations x + y + []z = 2 3x + y + z = 4 x + 2z = 1 have a unique…

Question

Let the system of linear equations

x + y + $$\alpha$$z = 2

3x + y + z = 4

x + 2z = 1

have a unique solution (x$^ *$, y$^ *$, z$^ *$). If ($$\alpha$$, x$^ *$), (y$^ *$, $$\alpha$$) and (x$^ *$, -y$^ *$) are collinear points, then the sum of absolute values of all possible values of $$\alpha$$ is

Accepted Answer

Given system of equations

$x + y + az = 2$ ..... (i)

$3x + y + z = 4$ ..... (ii)

$x + 2z = 1$ ..... (iii)

Solving (i), (ii) and (iii), we get

x = 1, y = 1, z = 0 (and for unique solution a $$\ne$$ -3)

Now, ($$\alpha$$, 1), (1, $$\alpha$$) and (1, -1) are collinear

$$\therefore$$ $$\left| {\begin{pmatrix} \alpha  & 1 & 1  \
   1 & \alpha  & 1  \
   1 & { - 1} & 1  \

\end{pmatrix} } \right| = 0$$

$$ \Rightarrow \alpha (\alpha  + 1) - 1(0) + 1( - 1 - \alpha ) = 0$$

$$ \Rightarrow {\alpha ^2} - 1 = 0$$

$$\therefore$$ $$\alpha  = \, \pm \,1$$

$$\therefore$$ Sum of absolute values of $$\alpha  = 1 + 1 = 2$$

Solutions · 1