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

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(A) The given system of equations is:$x + 2y - 3z = 1$$0x + 0y + (k + 3)z = 3$$(2k + 1)x + 0y + z = 0$The system is represented by the matrix equation

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$-3$

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(A) The given system of equations is:$x + 2y - 3z = 1$$0x + 0y + (k + 3)z = 3$$(2k + 1)x + 0y + z = 0$The system is represented by the matrix equation $AX = B$,where $A = \begin{bmatrix} 1 & 2 & -3 \ 0 & 0 & k+3 \ 2k+1 & 0 & 1 \end{bmatrix}$.For the system to be inconsistent,the determinant $D = |A|$ must be $0$,and at least one of the Cramer's rule determinants $(D_x, D_y, D_z)$ must be non-zero.First,calculate $D$:$D = 1(0 - 0) - 2(0 - (k+3)(2k+1)) - 3(0 - 0)$$D = 2(k+3)(2k+1)$Setting $D = 0$ gives $k = -3$ or $k = -1/2$.If $k = -3$,the second equation becomes $0 = 3$,which is a contradiction. Thus,the system is inconsistent for $k = -3$.If $k = -1/2$,the third equation becomes $0x + 0y + z = 0$,and the second equation becomes $2.5z = 3$,which is consistent.

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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