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 :
Step-by-Step Solution
Key Concept: A homogeneous system of linear equations AX = 0 has a unique solution (the trivial solution x=y=z=0) if and only if the determinant of the coefficient matrix A is non-zero.
The system of equations is:<br>x + ay + 0z = 0<br>0x + y + az = 0<br>ax + 0y + z = 0<br>The coefficient matrix is A = [[1, a, 0], [0, 1, a], [a, 0, 1]].<br>For a unique solution, det(A) \neq 0.<br>det(A) = 1(1 - 0) - a(0 - a^2) + 0 = 1 + a^3.<br>We require 1 + a^3 \neq 0, which means a^3 \neq -1, so a \neq -1.
Correct Answer: 2