The system of equations $-2x+y+z=a; x-2y+z=b; x+y-2z=c$ has unique solution, if :

Question

Accepted Answer

To determine why Option 4 is the correct answer, we need to analyze the conditions under which the given system of equations has a unique solution. The system of equations is:

1. $-2x + y + z = a$
2. $x - 2y + z = b$
3. $x + y - 2z = c$

For a system of linear equations to have a unique solution, the determinant of the coefficient matrix must be non-zero. The coefficient matrix for this system is:

$$
\begin{bmatrix}
-2 & 1 & 1 \
1 & -2 & 1 \
1 & 1 & -2
\end{bmatrix}
$$

The determinant of this matrix can be calculated as follows:

$$
\text{det} = (-2)((-2)(-2) - (1)(1)) - (1)((1)(-2) - (1)(1)) + (1)((1)(1) - (1)(-2))
$$

$$
= (-2)(4 - 1) - (1)(-2 - 1) + (1)(1 + 2)
$$

$$
= (-2)(3) - (1)(-3) + (1)(3)
$$

$$
= -6 + 3 + 3 = 0
$$

Since the determinant is zero, the coefficient matrix is singular, meaning the system does not have a unique solution. This confirms that Option 4, "Unique solution is not possible," is correct.

Now, let's examine why the other options are incorrect:

1. **Option 1: $a+b+c=0$**

This condition relates to the values of the constants $a$, $b$, and $c$, but it does not affect the determinant of the coefficient matrix. Since the determinant is zero, the system cannot have a unique solution regardless of the values of $a$, $b$, and $c$.

2. **Option 2: $a=b=c$**

Similar to Option 1, this condition specifies a relationship between the constants. However, it does not change the fact that the determinant of the coefficient matrix is zero. Therefore, the system cannot have a unique solution.

3. **Option 3: $a=2, b=2, c=-2$**

This option provides specific values for $a$, $b$, and $c$. Again, these values do not influence the determinant of the coefficient matrix. Since the determinant is zero, the system cannot have a unique solution.

In conclusion, the determinant of the coefficient matrix being zero is the key reason why the system does not have a unique solution, supporting Option 4 as the correct answer.