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(D) The system of equations is given by:$x + y + z = 6$$x + 2y + 3z = 10$$x + 2y + \lambda z = 12$$A$ system of linear equations $AX = B$ is inconsist

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$3$

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(D) The system of equations is given by:$x + y + z = 6$$x + 2y + 3z = 10$$x + 2y + \lambda z = 12$$A$ system of linear equations $AX = B$ is inconsistent if the determinant of the coefficient matrix $D = 0$ and at least one of the Cramer's rule determinants $(D_x, D_y, D_z)$ is non-zero.First,calculate the determinant $D$:$D = \begin{vmatrix} 1 & 1 & 1 \ 1 & 2 & 3 \ 1 & 2 & \lambda \end{vmatrix}$Expanding along the first row:$D = 1(2\lambda - 6) - 1(\lambda - 3) + 1(2 - 2)$$D = 2\lambda - 6 - \lambda + 3 + 0$$D = \lambda - 3$For the system to be inconsistent,we set $D = 0$:$\lambda - 3 = 0 \Rightarrow \lambda = 3$Now,check for consistency at $\lambda = 3$ by calculating $D_z$:$D_z = \begin{vmatrix} 1 & 1 & 6 \ 1 & 2 & 10 \ 1 & 2 & 12 \end{vmatrix}$$D_z = 1(24 - 20) - 1(12 - 10) + 6(2 - 2)$$D_z = 1(4) - 1(2) + 6(0) = 4 - 2 = 2$Since $D = 0$ and $D_z 
eq 0$,the system is inconsistent when $\lambda = 3$.

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 = $ ........

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