Given system of equations is homogeneous which is x+ay=0 y+az=0 z+ax=0 It can be written in matrix form as A=(
1
a
0
0
1
a
a
0
1
) Now, |A|=[1−a(−a2)]=1+a3≠0 So, system has only trivial solution. Now, |A|=0 only when a=−1 So, system of equations has infinitely many solutions which is not possible because it is given that system has a unique solution. Hence set of all real values of ca ' is R−{−1}