If the system of equations `3x+y+z=1, 6x+3y+2z=1` and `mux+lambday+3z=1` is inconsistent, then

Options

A. `mu ne 9, lambda ne 5`
B. `mu ne 9, lambda=5`
C. `mu=9, lambda=5`
D. `mu=9, lambda ne 5`

Answer

To determine the values of \( \mu \) and \( \lambda \) such that the system of equations is inconsistent, we will follow these steps:

### Step 1: Write down the equations
We have the following system of equations:
1. \( 3x + y + z = 1 \) (Equation 1)
2. \( 6x + 3y + 2z = 1 \) (Equation 2)
3. \( \mu x + \lambda y + 3z = 1 \) (Equation 3)

### Step 2: Form the coefficient matrix and calculate the determinant
The coefficient matrix for the system is:
\[ \begin{bmatrix} 3 & 1 & 1 \\ 6 & 3 & 2 \\ \mu & \lambda & 3 \end{bmatrix} \]

We denote the determinant of this matrix as \( \Delta \). For the system to be inconsistent, we need \( \Delta = 0 \).

### Step 3: Calculate \( \Delta \)
The determinant \( \Delta \) can be calculated using the formula for the determinant of a 3x3 matrix:
\[ \Delta = a(ei - fh) - b(di - fg) + c(dh - eg) \]
Substituting the values from our matrix:
\[ \Delta = 3(3 \cdot 3 - 2 \cdot \lambda) - 1(6 \cdot 3 - 2 \cdot \mu) + 1(6 \lambda - 3 \mu) \]
Calculating each term:
\[ = 3(9 - 2\lambda) - (18 - 2\mu) + (6\lambda - 3\mu) \]
\[ = 27 - 6\lambda - 18 + 2\mu + 6\lambda - 3\mu \]
\[ = 9 - \mu - \lambda \]

### Step 4: Set \( \Delta = 0 \)
For the system to be inconsistent:
\[ 9 - \mu - \lambda = 0 \]
This implies:
\[ \mu + \lambda = 9 \quad (1) \]

### Step 5: Calculate \( \Delta_1 \)
Next, we calculate \( \Delta_1 \) by replacing the first column with the constants from the right-hand side:
\[ \Delta_1 = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 3 & 2 \\ 1 & \lambda & 3 \end{vmatrix} \]
Calculating this determinant:
\[ = 1(3 \cdot 3 - 2 \cdot \lambda) - 1(1 \cdot 3 - 2 \cdot 1) + 1(1 \cdot \lambda - 3 \cdot 1) \]
\[ = 3 \cdot 3 - 2\lambda - (3 - 2) + (\lambda - 3) \]
\[ = 9 - 2\lambda - 1 + \lambda - 3 \]
\[ = 5 - \lambda \]

### Step 6: Calculate \( \Delta_2 \)
Now we calculate \( \Delta_2 \) by replacing the second column:
\[ \Delta_2 = \begin{vmatrix} 3 & 1 & 1 \\ 6 & 1 & 2 \\ \mu & 1 & 3 \end{vmatrix} \]
Calculating this determinant:
\[ = 3(1 \cdot 3 - 2 \cdot 1) - 1(6 \cdot 3 - 2 \mu) + 1(6 \cdot 1 - 1 \cdot \mu) \]
\[ = 3(3 - 2) - (18 - 2\mu) + (6 - \mu) \]
\[ = 3 - 18 + 2\mu + 6 - \mu \]
\[ = -9 + \mu \]

### Step 7: Calculate \( \Delta_3 \)
Finally, we calculate \( \Delta_3 \) by replacing the third column:
\[ \Delta_3 = \begin{vmatrix} 3 & 1 & 1 \\ 6 & 3 & 1 \\ \mu & \lambda & 1 \end{vmatrix} \]
Calculating this determinant:
\[ = 3(3 \cdot 1 - 1 \cdot \lambda) - 1(6 \cdot 1 - 1 \cdot \mu) + 1(6 \cdot \lambda - 3 \cdot \mu) \]
\[ = 3(3 - \lambda) - (6 - \mu) + (6\lambda - 3\mu) \]
\[ = 9 - 3\lambda - 6 + \mu + 6\lambda - 3\mu \]
\[ = 3 + 3\lambda - 2\mu \]

### Step 8: Conditions for inconsistency
For the system to be inconsistent, we need:
1. \( \Delta = 0 \) which gives us \( \mu + \lambda = 9 \)
2. At least one of \( \Delta_1, \Delta_2, \Delta_3 \) must be non-zero.

From the calculations:
- \( \Delta_1 = 5 - \lambda \)
- \( \Delta_2 = -9 + \mu \)
- \( \Delta_3 = 3 + 3\lambda - 2\mu \)

### Step 9: Solve for \( \mu \)
From \( \Delta_2 = 0 \):
\[ -9 + \mu = 0 \implies \mu = 9 \]

### Step 10: Substitute \( \mu \) into \( (1) \)
Substituting \( \mu = 9 \) into \( \mu + \lambda = 9 \):
\[ 9 + \lambda = 9 \implies \lambda = 0 \]

### Conclusion
Thus, the values of \( \mu \) and \( \lambda \) that make the system inconsistent are:
\[ \mu = 9, \quad \lambda = 0 \]