Question
If the system of equations $$x + ay = 0, az + y = 0$$ and $$ax + z = 0$$ has infinite solutions, then the value of $$a$$ is
A.
$$- 1$$
B.
1
C.
0
D.
no real values
Answer :
$$- 1$$
Solution :
The given system is, $$x + ay = 0$$
$$az + y = 0$$
$$ax + z = 0$$
It is system of homogeneous equations therefore, it will have infinite many solutions if determinant of co-efficient matrix is zero. i.e.,
\[\left| \begin{array}{l}
1\,\,\,\,\,\,\,a\,\,\,\,\,\,\,0\\
0\,\,\,\,\,\,\,1\,\,\,\,\,\,\,a\\
a\,\,\,\,\,\,\,0\,\,\,\,\,\,\,1
\end{array} \right| = 0\]
$$\eqalign{
& \Rightarrow \,\,1\left( {1 - 0} \right) - a\left( {0 - {a^2}} \right) = 0 \cr
& \Rightarrow \,\,1 + {a^3} = 0 \cr
& \Rightarrow \,\,{a^3} = - 1 \cr
& \Rightarrow \,\,a = - 1 \cr} $$