Question

Consider the system of equations :
x + ay = 0, y + az = 0 and z + ax = 0. Then the set of all real values of ‘a’ for which the system has a unique solution is:

• A. R - {1}
• B. R - { - 1}
• C. {1, - 1}
• D. {1, 0, -1}

Answer 1

Answer: (B)

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 = \begin{pmatrix}1&a&0\\ 0&1&a\\ a&0&1\end{pmatrix} $
Now, $| A | = [1 - a(- a^2)] = 1 + a^3 \ne 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 ‘a’ is R - {- 1}.