Matrix theory can be aplied to investigate the conditions under which a given system of linear equations possesses unique, indinite or no solutions. Consider the system of non-homogeneous linear equations in 3 unknowns
`x+y+x=6`
`x+2y+3z=10`
`x+2y+lamdaz=rho`
and answer the questions that follow.
The systme possesses a unique solution if

Options

A. `lamda ne 3`
B. `lamda ne 3, rho ne 10`
C. `rho ne 10`
D. `rho = 10`

Answer

To determine the conditions under which the given system of non-homogeneous linear equations possesses a unique solution, we need to analyze the determinant of the coefficients of the variables in the equations.

The given equations are:
1. \( x + y + z = 6 \)
2. \( x + 2y + 3z = 10 \)
3. \( x + 2y + \lambda z = \rho \)

### Step 1: Write the Coefficient Matrix
We can write the coefficient matrix \( A \) from the coefficients of \( x, y, z \) in the equations:

\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{bmatrix} \]

### Step 2: Calculate the Determinant
To find the condition for a unique solution, we need to calculate the determinant \( D \) of the matrix \( A \). The system has a unique solution if \( D \neq 0 \).

The determinant \( D \) can be calculated as follows:

\[ D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{vmatrix} \]

Using the determinant formula for a 3x3 matrix, we have:

\[ D = 1 \cdot \begin{vmatrix} 2 & 3 \\ 2 & \lambda \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 3 \\ 1 & \lambda \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 2 \\ 1 & 2 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. \( \begin{vmatrix} 2 & 3 \\ 2 & \lambda \end{vmatrix} = 2\lambda - 6 \)

2. \( \begin{vmatrix} 1 & 3 \\ 1 & \lambda \end{vmatrix} = \lambda - 3 \)

3. \( \begin{vmatrix} 1 & 2 \\ 1 & 2 \end{vmatrix} = 0 \)

Putting these back into the determinant formula:

\[ D = 1(2\lambda - 6) - 1(\lambda - 3) + 0 \]
\[ D = 2\lambda - 6 - \lambda + 3 \]
\[ D = \lambda - 3 \]

### Step 3: Set the Condition for Unique Solution
For the system to have a unique solution, we require:

\[ D \neq 0 \implies \lambda - 3 \neq 0 \implies \lambda \neq 3 \]

### Conclusion
The system of equations possesses a unique solution if \( \lambda \neq 3 \).

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