Question

The value of $\lambda$ such that the system $x - 2y + z = -4, 2x - y + 2z = 2$ and $x + y + \lambda z = 4$ has no solution is

• A. 3
• B. 1
• C. 0
• D. -3

Answer 1

Answer: (B)

1

Answer 2

For the system

$$\begin{cases} x - 2y + z = -4,\\[6pt] 2x - y + 2z = 2,\\[6pt] x + y + \lambda z = 4, \end{cases}$$

the coefficient matrix is

$$A = \begin{pmatrix} 1 & -2 & 1 \\ 2 & -1 & 2 \\ 1 & 1 & \lambda \end{pmatrix}.$$

1. Compute the determinant:

   $$\det A = 1\begin{vmatrix} -1 & 2 \\ 1 & \lambda \end{vmatrix} - (-2)\begin{vmatrix} 2 & 2 \\ 1 & \lambda \end{vmatrix} + 1\begin{vmatrix} 2 & -1 \\ 1 & 1 \end{vmatrix}.$$ $$= 1(-\lambda - 2) + 2(2\lambda - 2) + 1(2+1)$$ $$= (-\lambda - 2) + (4\lambda - 4) + 3$$ $$= 3\lambda - 3.$$

   Setting $\det A = 0$ gives:

   $$3\lambda - 3 = 0\quad\Rightarrow\quad \lambda = 1.$$

2. Check inconsistency:

   With $\lambda = 1$, the third equation should be a linear combination of the first two for consistency. Testing the combination shows that while the coefficients match, the corresponding constant term does not (combining the first two gives a constant 6 instead of 4). Hence, the system is inconsistent (has no solution).