find the value of “a” such that the following system of linear 

equations 

𝒙 + 𝒚 + 𝒂𝒛 = 𝟑, 𝒙 + 𝒂𝒚 + 𝒛 = 𝟐, 𝒂𝒙 + 𝒚 − 𝒛 = 𝟏

have

i) exactly one solution

ii) No solution

iii) Infinite number of solutions and find them

To analyze the conditions for the system of equations to have exactly one solution, no solution, or infinitely many solutions, we’ll start by representing the system in matrix form and determining when it is consistent, consistent with a unique solution, or inconsistent.

### System of Equations

The system is:

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

In matrix form, this can be written as:
\[
\begin{pmatrix} 1 & 1 & a \\ 1 & a & 1 \\ a & 1 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}
\]

To determine the solution types, we’ll examine the determinant of the coefficient matrix.

### Step 1: Calculate the Determinant

The determinant \( D \) of the matrix is:
\[
D = \begin{vmatrix} 1 & 1 & a \\ 1 & a & 1 \\ a & 1 & -1 \end{vmatrix}
\]

Expanding along the first row, we get:
\[
D = 1 \cdot \begin{vmatrix} a & 1 \\ 1 & -1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 1 \\ a & -1 \end{vmatrix} + a \cdot \begin{vmatrix} 1 & a \\ a & 1 \end{vmatrix}
\]

Calculating each minor:
1. \( \begin{vmatrix} a & 1 \\ 1 & -1 \end{vmatrix} = a(-1) - 1 \cdot 1 = -a - 1 = -(a + 1) \)
2. \( \begin{vmatrix} 1 & 1 \\ a & -1 \end{vmatrix} = 1(-1) - 1 \cdot a = -1 - a = -(1 + a) \)
3. \( \begin{vmatrix} 1 & a \\ a & 1 \end{vmatrix} = 1 \cdot 1 - a \cdot a = 1 - a^2 \)

Thus:
\[
D = 1 \cdot (-(a + 1)) - 1 \cdot (-(1 + a)) + a \cdot (1 - a^2)
\]
\[
D = -(a + 1) + (1 + a) + a(1 - a^2)
\]
\[
D = 0 + a(1 - a^2)
\]


Find the value of 'a' such that the following system of linear equations has: (i) exactly one solution, (ii) no solution, (iii) an infinite number of solutions, and find them.

x + y + az = 3
x + ay + z = 2
a*x + y - z = 1

Explore how to determine the values of 'a' for a system of linear equations to have exactly one solution, no solution, or infinitely many solutions. This problem covers essential linear algebra concepts, including matrix determinants and conditions for consistency in systems of equations.

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