For what value of λ the given system of equations x+y+z=6x+2y+3z=10x+2y+λz=12 is inconsistent?

The given system of equations may be written as 1 1 11 2 31 2 λ xyz=61012⇒11101201λ−1xyz=646 Applying R2→R2−R1R3→R3−R1⇒11101200λ−3xyz=642 Applying R3→R3−R2 For λ=3, we observe that rank of matrix A is 2 and that of the augmented matrix is 3. So, the system is inconsistent.

For what value of λ the given system of equations x+y+z=6x+2y+3z=10x+2y+λz=12 is inconsistent?

The given system of equations may be written as 1 1 11 2 31 2 λ xyz=61012⇒11101201λ−1xyz=646 Applying R2→R2−R1R3→R3−R1⇒11101200λ−3xyz=642 Applying R3→R3−R2 For λ=3, we observe that rank of matrix A is 2 and that of the augmented matrix is 3. So, the system is inconsistent.

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For what value of $λ$ the given system of equations $\begin{array}{c} x + y + z = 6 \\ x + 2 y + 3 z = 10 \\ x + 2 y + λ z = 12 \end{array}$ is inconsistent?

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$λ = 1$

$λ = 2$

$λ = - 2$

$λ = 3$

answer is D.

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Detailed Solution

The given system of equations may be written as

$[\begin{array}{lll} 1 1 1 \\ 1 2 3 \\ 1 2 λ \end{array}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{matrix} 6 \\ 10 \\ 12 \end{matrix}]$

$\Rightarrow [\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 1 & λ - 1 \end{array}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{array}{l} 6 \\ 4 \\ 6 \end{array}]$ Applying $\begin{aligned} R_{2} \to R_{2} - R_{1} \\ R_{3} \to R_{3} - R_{1} \end{aligned}$

$\Rightarrow [\begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & λ - 3 \end{array}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{array}{l} 6 \\ 4 \\ 2 \end{array}]$ Applying $R_{3} \to R_{3} - R_{2}$