Discussion
The following system of linear equations
\(7 x - 3 y + z = 0\)
\(3 x - y + z = 0\)
\(x - y - z = 0\)
has:
Answer: A
The system has infinitely many solutions.
It is homogeneous (every right-hand side is 0), so \(x = y = z = 0\) always works. It has other solutions as well exactly when the coefficient determinant is zero:
\[ \begin{aligned} \Delta &= 7(1 + 1) + 3(-3 - 1) + 1(-3 + 1) \\ &= 14 - 12 - 2 = 0 \end{aligned} \]
The rank is 2 (the first two rows are not multiples of each other), so the solutions form a line through the origin: for example \(x = 1, y = 2, z = -1\) and any multiple of it.
A unique solution would need \(\Delta \neq 0\). A homogeneous system can never have no solution, and a linear system cannot have exactly three.
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