Question

The non-trivial solutions of the equations:
\[ x + y - 6z = 0 \] \[ -3x + y + 2z = 0 \] \[ x - y + 2z = 0 \]

• A. $x = 2c, y = 4c, z = 3c, c \neq 0$ is any scalar.
• B. $x = 2c, y = 4c, z = c, c \neq 0$ is any scalar.
• C. $x = 2c, y = 4c, z = 2c, c \neq 0$ is any scalar.
• D. $x = c, y = 4c, z = 2c, c \neq 0$ is any scalar.

Hint:

To quickly check options in competitive exams, substitute $x=2c, y=4c, z=c$ directly into the equations: $2c + 4c - 6c = 0 \quad \checkmark$ $-3(2c) + 4c + 2c = 0 \quad \checkmark$ $2c - 4c + 2c = 0 \quad \checkmark$ This confirms the result rapidly without doing full matrix reduction.

Answer 1

The Correct Option is B