Question

Let $\lambda \in R.$ the system of linear equations

$\begin{array}{rcl}2x_1-4x_2+\lambda x_3& =& 1\\ x_1-6x_2+x_3& =& 2\\ \lambda x_1-10x_2+4x_3& =& 3\end{array}$

is inconsistent for:

• A. exactly two values of$\lambda$
• B. exactly one negative value of$\lambda .$
• C. every value of $\lambda .$
• D. exactly one positive value of $\lambda .$

Answer 1

The correct option is B

exactly one negative value of$\lambda .$

Explanation for the correct answer:

Identifying the value of inconsistency for the given equations:

Given equations are

$\begin{array}{rcl}2x_1-4x_2+\lambda x_3& =& 1\\ x_1-6x_2+x_3& =& 2\\ \lambda x_1-10x_2+4x_3& =& 3\end{array}$

Let's frame the matrix from the above equations

$\begin{array}{rcl}D& =& \left|\begin{array}{ccc}2& -4& \lambda \\ 1& -6& 1\\ \lambda & -10& 4\end{array}\right|\\ & =& 2(3\lambda +2)(\lambda -3)\end{array}$

Thus,

$\begin{array}{rcl}D1& =& -2(\lambda -3)\\ D2& =& -2(\lambda +1)(\lambda -3)\\ D3& =& –2(\lambda -3)\\ \text{When}\lambda & =& 3,\text{then}\\ D& =& D1=D2=D3=0\end{array}$

Hence, infinitely many solutions

$\begin{array}{rcl}\text{When}\lambda & =& \frac{–2}{3}\text{then}D1,D2,D3\text{none of them is zero so equations are inconsistent}\\ \lambda & =& \frac{–2}{3}\end{array}$

Hence the correct answer is option (B).