For what value of λ, the system of equations

$x+y+z = 6$

$x+2y+ 3z=10$

$x + 2y + λz=12$

is inconsistent?

$λ=3$

The given system of equations may be written as

$\begin{bmatrix}1&1&1\\1&2&3\\1&2&λ\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}6\\10\\12\end{bmatrix}$

$⇒\begin{bmatrix}1&1&1\\0&1&2\\0&1&λ-1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}6\\4\\6\end{bmatrix}$ Applying $R_2 → R_2 - R_1, R_3→R_3-R_1$

$⇒\begin{bmatrix}1&1&1\\0&1&2\\0&0&λ-3\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}6\\4\\2\end{bmatrix}$  Applying $R_3 → R_3-R_2$

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.