Classify the following system of equations as consistent or inconsistent :
`{:(x+y+z=1),(2x+3y+2z=2),(ax+ay+2az=4):}`

Answer

To classify the given system of equations as consistent or inconsistent, we will follow these steps:

### Step 1: Write the system of equations in matrix form
The given equations are:
1. \( x + y + z = 1 \)
2. \( 2x + 3y + 2z = 2 \)
3. \( ax + ay + 2az = 4 \)

We can express this system in the form \( Ax = b \), where:
- \( A \) is the coefficient matrix,
- \( x \) is the variable matrix,
- \( b \) is the constant matrix.

The coefficient matrix \( A \) is:
\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ a & a & 2a \end{pmatrix} \]

The variable matrix \( x \) is:
\[ x = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \]

The constant matrix \( b \) is:
\[ b = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} \]

### Step 2: Calculate the determinant of the coefficient matrix \( A \)
To determine if the system is consistent or inconsistent, we need to calculate the determinant of matrix \( A \).

The determinant of \( A \) can be calculated using the formula for the determinant of a 3x3 matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix:
\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ a & a & 2a \end{pmatrix} \]

Calculating the determinant:
\[ \text{det}(A) = 1 \cdot (3 \cdot 2a - 2 \cdot a) - 1 \cdot (2 \cdot 2a - 2 \cdot a) + 1 \cdot (2 \cdot a - 3 \cdot a) \]
\[ = 1 \cdot (6a - 2a) - 1 \cdot (4a - 2a) + 1 \cdot (2a - 3a) \]
\[ = 1 \cdot 4a - 1 \cdot 2a - 1 \cdot a \]
\[ = 4a - 2a - a = a \]

### Step 3: Classify the system based on the determinant
Now, we analyze the determinant:
- If \( \text{det}(A) = 0 \), the system is inconsistent.
- If \( \text{det}(A) \neq 0 \), the system is consistent.

From our calculation, we found that:
\[ \text{det}(A) = a \]

Thus:
- If \( a = 0 \), the system is inconsistent.
- If \( a \neq 0 \), the system is consistent.

### Conclusion
The classification of the system of equations depends on the value of \( a \):
- The system is **inconsistent** if \( a = 0 \).
- The system is **consistent** if \( a \neq 0 \).