We are given the system

with $a\in \mathbb R\setminus\{0\}$.

Let

• $S_1$ be the set of all $a$ for which the system has a unique solution,
• $S_2$ be the set of all $a$ for which the system has infinitely many solutions.

We must determine the correct option.

1. Coefficient matrix

The coefficient matrix is

A system of 3 linear equations in 3 unknowns has:

1. a unique solution if $\det(A)\neq 0$,
2. infinitely many solutions if $\det(A)=0$ and the system is consistent with rank $<3$.

So first we compute $\det(A)$.

2. Compute the determinant

Expand along the first row:

Now compute each minor.

First minor

Second minor

Third minor

Substitute back:

3. When is the determinant zero?

Since $a\in \mathbb R\setminus\{0\}$, we only need to check

Its discriminant is

So the quadratic has no real root.

Therefore, for every real $a\neq 0$,

Hence the system has a unique solution for every $a\in \mathbb R\setminus\{0\}$.

So,

4. Determine $S_2$

For infinitely many solutions, we need $\det(A)=0$ first. But we found that for all real $a\neq 0$,

Thus there is no value of $a\in \mathbb R\setminus\{0\}$ for which the system has infinitely many solutions.

So,

5. Match with the options

We obtained:

This matches Option C.

Final Answer