The system of linear equations 3x - 2y - kz = | vedclass

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Question

(D) For the system to be inconsistent,the determinant of the coefficient matrix $\Delta$ must be $0$,and at least one of $\Delta_x, \Delta_y, \Delta_z

Vedclass · Accepted Answer

$k = 3, m 
eq \frac{4}{5}$

Vedclass · Answer

(D) For the system to be inconsistent,the determinant of the coefficient matrix $\Delta$ must be $0$,and at least one of $\Delta_x, \Delta_y, \Delta_z$ must be non-zero.First,calculate $\Delta = \begin{vmatrix} 3 & -2 & -k \ 2 & -4 & -2 \ 1 & 2 & -1 \end{vmatrix} = 3(4 + 4) + 2(-2 + 2) - k(4 + 4) = 24 - 8k$.Setting $\Delta = 0$ gives $24 - 8k = 0$,so $k = 3$.Now,check $\Delta_z$ with $k = 3$:$\Delta_z = \begin{vmatrix} 3 & -2 & 10 \ 2 & -4 & 6 \ 1 & 2 & 5m \end{vmatrix} = 3(-20m - 12) + 2(10m - 6) + 10(4 + 4) = -60m - 36 + 20m - 12 + 80 = -40m + 32$.For inconsistency,$\Delta_z 
eq 0$,so $-40m + 32 
eq 0 \Rightarrow 40m 
eq 32 \Rightarrow m 
eq \frac{32}{40} \Rightarrow m 
eq \frac{4}{5}$.Thus,the system is inconsistent if $k = 3$ and $m 
eq \frac{4}{5}$.

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

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