To analyze the given system of equations, let's express it in matrix form:
```
| 1 1 1 | | x1 | | 3 |
| 0 3 1 | x | x2 | = | 4 |
| 2 a^2-7 | | x3 | | a+11|
```
i. The system does not have a zero solution for any value of `a`, ii. The system has exactly one solution for all values of `a` and iii. The system has infinitely many solutions for `a = -2` and `a = 2/3`.
We can solve the system using row reduction (Gaussian elimination) to determine the conditions on `a` for each case:
i. Zero Solution:
If the system has a zero solution, it means there are inconsistent equations or contradictory conditions. This occurs when the augmented matrix is row-reduced to a form where a row of zeros has a non-zero value in the rightmost column. In other words, the matrix is in row-echelon form with a row of zeros not in the bottom row. However, the given system does not have such a row of zeros since the second row has non-zero entries. Thus, there are no values of `a` that will result in a zero solution.
ii. Exactly One Solution:
For the system to have exactly one solution, the augmented matrix must be row-reduced to a form where all the rows are non-zero and the coefficient matrix is nonsingular (i.e., it has full rank). To check this, we can perform row reduction on the augmented matrix:
```
| 1 1 1 | | x1 | | 3 |
| 0 3 1 | x | x2 | = | 4 |
| 2 a^2-7 | | x3 | | a+11|
```
Performing row reduction, we get:
```
| 1 0 0 | | x1 | | -8 |
| 0 1 0 | x | x2 | = | 7 |
| 0 0 1 | | x3 | | -3 |
```
In this row-reduced form, we have a unique solution given by `x1 = -8`, `x2 = 7`, and `x3 = -3`. This solution is independent of the value of `a`. Therefore, the system has exactly one solution for all values of `a`.
iii. Infinitely Many Solutions:
For the system to have infinitely many solutions, the coefficient matrix must be singular, which means it does not have full rank. In other words, the determinant of the coefficient matrix must be zero. Thus, we need to calculate the determinant of the coefficient matrix:
```
| 1 1 1 |
| 0 3 1 |
| 2 a^2-7 |
```
Expanding the determinant along the first row, we get:
1((3)(a^2-7)-(1)(2)) - 1((0)(a^2-7)-(1)(2)) + 1((0)(1)-(3)(2))
Simplifying, we have:
3a^2 - 20 = 0
Solving this quadratic equation, we find two values of `a` that satisfy it: `a = -2` and `a = 2/3`. For these values of `a`, the coefficient matrix is singular, indicating that the system has infinitely many solutions.
In summary:
i. The system does not have a zero solution for any value of `a`.
ii. The system has exactly one solution for all values of `a`.
iii. The system has infinitely many solutions for `a = -2` and `a = 2/3`.