If $a, b, c$ are non-zero real numbers and if the system of equations $(a – 1 )x = y + z,$  $(b – 1 )y = z + x ,$ $(c – 1 )z= x + y,$ has a non-trivial solution, then $ab + bc + ca$ equals

If the system of equations  $x +y + z = 6$ ; $x + 2y + 3z= 10$ ; $x + 2y + \lambda z = 0$ has a unique solution, then $\lambda $ is not equal to

Let $\mathrm{A}$ be a square matrix of order $3 \times 3$ , then $|\mathrm{k A}|$ is equal to

If $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = k(a + b + c)({a^2} + {b^2} + {c^2}$ $ – bc – ca – ab)$, then $k =$

The system of linear equations  $3 x-2 y-k z=10$; $2 x-4 y-2 z=6$ ; $x+2 y-z=5\, m$ is inconsistent if