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To determine the values of \( a \) for which the given system has no solution, a unique solution, or infinitely many solutions, we need to analyze the coefficients of the system and how they change with \( a \). The system is given as:

1. \( ax + y + z = -2 \)
2. \( x + ay + z = 1 \)
3. \( x + y + az = 1 \)

We can form the coefficient matrix \( A \) and use its determinant to check the conditions for different types of solutions:

\[ A = \begin{bmatrix} a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & a \end{bmatrix} \]

### Determinant of A

The determinant of matrix \( A \), denoted as \( \det(A) \), is calculated as follows:

\[ \det(A) = a(a^2 - 2) + 1(-1 + 1) + 1(1 - a) = a^3 - 2a + 1 - a \]

\[ \det(A) = a^3 - 3a + 1 \]

We solve for the roots of \( \det(A) = 0 \) to find the values of \( a \) that might lead to special cases:

\[ a^3 - 3a + 1 = 0 \]

Let's solve this cubic equation to find the potential values of \( a \).The roots of the equation \( a^3 - 3a + 1 = 0 \) are complex and can be simplified numerically for easier interpretation. Let's find the numerical values of these roots.The numerical solutions for \( a \) where \( \det(A) = 0 \) are approximately:

1. \( a \approx 0.347 \)
2. \( a \approx 1.532 \)
3. \( a \approx -1.879 \)

### Analyzing the Types of Solutions

**a) No Solution:** Typically, a system of equations has no solution if there is a contradiction, such as parallel planes or a row of zeroes in the augmented matrix corresponding to a nonzero constant. This occurs when the determinant of the coefficient matrix is zero, and the corresponding augmented matrix determinant does not match this.

**b) A Unique Solution:** The system will have a unique solution if \( \det(A) \neq 0 \), meaning \( a \

Consider the following linear system in 3 variables:
ax + y + z = -2
x + ay + z = 1
x + y + az = 1
Find all values of a ∈ R such that the system has
(a) no solution.
(b) a unique solution.
(c) infinitely many solutions. In each case, find the solution set.

Solve a 3-variable system and determine the values of 'a' such that the system has no solution, a unique solution, or infinitely many solutions. Includes detailed analysis using determinants and algebraic properties.

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