If the system of equations x + 2y - 3z = 1,(k | vedclass

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(A) system of linear equations is inconsistent if the determinant of the coefficient matrix $D = 0$ and at least one of the Cramer's rule determinants

Vedclass · Accepted Answer

$-3$

Vedclass · Answer

(A) system of linear equations 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.The coefficient matrix is:$D = \begin{vmatrix} 1 & 2 & -3 \ 0 & 0 & k+3 \ 2k+1 & 0 & 1 \end{vmatrix}$Expanding along the second column:$D = -2 \begin{vmatrix} 0 & k+3 \ 2k+1 & 1 \end{vmatrix} = -2(0 - (k+3)(2k+1)) = 2(k+3)(2k+1)$Setting $D = 0$ gives $k = -3$ or $k = -1/2$.Case $1$: If $k = -3$,the second equation becomes $0 = 3$,which is a contradiction. Thus,the system is inconsistent.Case $2$: If $k = -1/2$,the equations are $x + 2y - 3z = 1$,$2.5z = 3$,and $z = 0$. This leads to $0 = 3$,which is also inconsistent. However,checking the options,$-3$ is provided.

If the system of equations $x + 2y - 3z = 1$,$(k + 3)z = 3$,and $(2k + 1)x + z = 0$ is inconsistent,then the value of $k$ is:

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