Let `lambda in R `. The system of linear equations
`2x_(1) - 4x_(2) + lambda x_(3) = 1`
`x_(1) -6x_(2) + x_(3) = 2`
`lambda x_(1) - 10x_(2) + 4 x_(3) = 3`
is inconsistent for :

Options

A. exactly one negative value of `lambda`
B. exactly one positive value of `lambda`
C. every value of `lambda`
D. exactly two values of `lambda`

Answer

To determine the values of \( \lambda \) for which the given system of linear equations is inconsistent, we need to analyze the system of equations:

1. \( 2x_1 - 4x_2 + \lambda x_3 = 1 \)
2. \( x_1 - 6x_2 + x_3 = 2 \)
3. \( \lambda x_1 - 10x_2 + 4x_3 = 3 \)

### Step 1: Formulate the Coefficient Matrix and Find the Determinant

The coefficient matrix \( A \) for the system is:

\[ A = \begin{bmatrix} 2 & -4 & \lambda \\ 1 & -6 & 1 \\ \lambda & -10 & 4 \end{bmatrix} \]

To find when the system is inconsistent, we need to compute the determinant of this matrix and set it equal to zero:

\[ \text{det}(A) = \begin{vmatrix} 2 & -4 & \lambda \\ 1 & -6 & 1 \\ \lambda & -10 & 4 \end{vmatrix} \]

### Step 2: Calculate the Determinant

Using the formula for the determinant of a \( 3 \times 3 \) matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

We can compute:

\[ \text{det}(A) = 2((-6)(4) - (1)(-10)) - (-4)((1)(4) - (1)(\lambda)) + \lambda((1)(-10) - (-6)(\lambda)) \]

Calculating each term:

1. \( 2((-24) + 10) = 2(-14) = -28 \)
2. \( -4(4 - \lambda) = -16 + 4\lambda \)
3. \( \lambda(-10 + 6\lambda) = -10\lambda + 6\lambda^2 \)

Putting it all together:

\[ \text{det}(A) = -28 + 4\lambda - 16 + 4\lambda + 6\lambda^2 \]
\[ = 6\lambda^2 + 8\lambda - 44 \]

### Step 3: Set the Determinant to Zero

Now, we set the determinant equal to zero:

\[ 6\lambda^2 + 8\lambda - 44 = 0 \]

### Step 4: Solve the Quadratic Equation

Dividing the entire equation by 2 for simplicity:

\[ 3\lambda^2 + 4\lambda - 22 = 0 \]

Using the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ \lambda = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 3 \cdot (-22)}}{2 \cdot 3} \]
\[ = \frac{-4 \pm \sqrt{16 + 264}}{6} \]
\[ = \frac{-4 \pm \sqrt{280}}{6} \]
\[ = \frac{-4 \pm 2\sqrt{70}}{6} \]
\[ = \frac{-2 \pm \sqrt{70}}{3} \]

### Step 5: Identify Values of \( \lambda \)

The solutions for \( \lambda \) are:

\[ \lambda_1 = \frac{-2 + \sqrt{70}}{3}, \quad \lambda_2 = \frac{-2 - \sqrt{70}}{3} \]

### Conclusion

The system of equations is inconsistent for the values of \( \lambda \) that satisfy the determinant being zero, which are:

\[ \lambda = \frac{-2 + \sqrt{70}}{3} \quad \text{and} \quad \lambda = \frac{-2 - \sqrt{70}}{3} \]