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(A) The system of equations is consistent if the determinant of the coefficient matrix $D$ is non-zero (unique solution) or if $D=0$ and the augmented

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$R$

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(A) The system of equations is consistent if the determinant of the coefficient matrix $D$ is non-zero (unique solution) or if $D=0$ and the augmented matrix satisfies the consistency condition.The determinant $D$ is given by:$D = \begin{vmatrix} 1 & 1 & 1 \ 3 & 2 & 5 \ 9 & 4 & 28+[\lambda] \end{vmatrix}$Expanding along the first row:$D = 1(2(28+[\lambda]) - 20) - 1(3(28+[\lambda]) - 45) + 1(12 - 18)$$D = (56 + 2[\lambda] - 20) - (84 + 3[\lambda] - 45) - 6$$D = (36 + 2[\lambda]) - (39 + 3[\lambda]) - 6$$D = -[\lambda] - 9$If $D 
eq 0$,i.e.,$[\lambda] 
eq -9$,the system has a unique solution.If $D = 0$,i.e.,$[\lambda] = -9$,we check for consistency using Cramer's rule or row reduction.For $[\lambda] = -9$,the third equation becomes $9x + 4y + 19z = -9$.The system is consistent if the augmented matrix has the same rank as the coefficient matrix. Since the system is consistent for all $[\lambda] \in \mathbb{Z}$,and $[\lambda]$ can take any integer value,the system has a solution for all $\lambda \in R$.

Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4$,$3x+2y+5z=3$,$9x+4y+(28+[\lambda])z=[\lambda]$ has a solution is:

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