Consider the system of equations
x+y+z=5, x+2y+3z=9, x+3y+`lambda z=mu`
The sustem is called smart brilliant good and lazy according as it has solution unique solution infinitely many solution respectively .
The system is smart if

Options

A. `lambdane5 "or" lambda=5 "and" mu =13`
B. `lambdane5 `
C. `lambdane5 " ""and"" " mu ne13`
D. `lambdane5 or lambda" ""and"" " mu ne13`

Answer

To determine the conditions under which the given system of equations has a unique solution, we will follow these steps:

### Step 1: Write the system of equations in matrix form

The given equations are:
1. \( x + y + z = 5 \)
2. \( x + 2y + 3z = 9 \)
3. \( x + 3y + \lambda z = \mu \)

We can express this system in matrix form \( A \mathbf{x} = \mathbf{b} \), where:
\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 9 \\ \mu \end{pmatrix} \]

### Step 2: Find the determinant of the coefficient matrix \( A \)

To find the condition for a unique solution, we need to calculate the determinant of matrix \( A \):
\[ \text{det}(A) = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda \end{vmatrix} \]

Using the determinant formula for a 3x3 matrix:
\[ \text{det}(A) = 1 \cdot (2 \cdot \lambda - 3 \cdot 3) - 1 \cdot (1 \cdot \lambda - 1 \cdot 3) + 1 \cdot (1 \cdot 3 - 1 \cdot 2) \]

Calculating this step-by-step:
1. \( 1 \cdot (2\lambda - 9) \)
2. \( -1 \cdot (\lambda - 3) \)
3. \( +1 \cdot (3 - 2) \)

Combining these:
\[ \text{det}(A) = 2\lambda - 9 - \lambda + 3 + 1 = \lambda - 5 \]

### Step 3: Set the determinant condition for a unique solution

For the system to have a unique solution, the determinant must not be equal to zero:
\[ \lambda - 5 \neq 0 \]
Thus, we find:
\[ \lambda \neq 5 \]

### Conclusion

The system of equations is called "smart" (has a unique solution) if:
\[ \lambda \neq 5 \]
The value of \( \mu \) can be any real number.