Question

If the system of linear equations
2 x + 3 y - z = -2
x + y + z = 4
x - y + |λ|z = 4λ - 4
where λ ∈ R, has no solution, then

• A. λ = 7
• B. λ = -7
• C. λ = 8
• D. λ² = 1

Answer 1

Answer: (B)

λ = -7

Answer 2

The correct answer is (B) : λ = -7
$\Delta = \left| \begin{array}{ccc} 2 & 3 & -1 \\\ 1 & 1 & 1 \\\ 1 & -1 & | \lambda | \end{array} \right| = 0$
$⇒ |λ| = 7$
But at λ = 7, D x = D y = D z = 0
P 1 : 2 x + 3 y – z = –2
P 2 :x + y + z = 4
P 3 :x – y + |λ|z = 4λ – 4
So clearly, 5 P 2 – 2 P 1 = P 3, so at λ = 7, system of equation is having infinite solutions.
Therefore, λ = –7 is correct answer.