If the system of equations $3x - 2y + z = 0$,$\lambda x - 14y + 15z = 0$,and $x + 2y + 3z = 0$ has a non-trivial solution,then $\lambda = $

Consider the system of equations:
$x - 2y + 3z = -1$; $-x + y - 2z = k$; $x - 3y + 4z = 1$
$STATEMENT-1$: The system of equations has no solution for $k \neq 3$.
$STATEMENT-2$: The determinant $\left|\begin{array}{ccc}1 & -2 & 3 \\ -1 & 1 & -2 \\ 1 & -3 & 4\end{array}\right| = 0$.

Let $A$ and $B$ be two $3 \times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then:

Solve the system of linear equations using the matrix method:
$5x + 2y = 3$
$3x + 2y = 5$

Consider the system of equations: $ax + by + cz = 2$,$bx + cy + az = 2$,$cx + ay + bz = 2$,where $a, b, c$ are real numbers such that $a + b + c = 0$. Then,the system

For what value of $\lambda$,the system of equations $x + y + z = 6$,$x + 2y + 3z = 10$,and $x + 2y + \lambda z = 12$ is inconsistent? $\lambda = $ ........