Let the system of linear equations x + 2y + z | vedclass

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(C) For a system of linear equations to be inconsistent,the determinant of the coefficient matrix $\Delta$ must be $0$,and at least one of the Cramer'

Vedclass · Accepted Answer

$-\frac{7}{2}$

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(C) For a system of linear equations to be inconsistent,the determinant of the coefficient matrix $\Delta$ must be $0$,and at least one of the Cramer's rule determinants $(\Delta_x, \Delta_y, \Delta_z)$ must be non-zero.The coefficient matrix is $A = \begin{bmatrix} 1 & 2 & 1 \ \alpha & 3 & -1 \ -\alpha & 1 & 2 \end{bmatrix}$.Calculating the determinant $\Delta$:$\Delta = 1(3 	imes 2 - (-1) 	imes 1) - 2(\alpha 	imes 2 - (-1) 	imes (-\alpha)) + 1(\alpha 	imes 1 - 3 	imes (-\alpha))$$\Delta = 1(6 + 1) - 2(2\alpha - \alpha) + 1(\alpha + 3\alpha)$$\Delta = 7 - 2\alpha + 4\alpha = 7 + 2\alpha$.Setting $\Delta = 0$ for inconsistency:$7 + 2\alpha = 0 \Rightarrow \alpha = -\frac{7}{2}$.Now,check $\Delta_x$ at $\alpha = -\frac{7}{2}$:$\Delta_x = \begin{vmatrix} 2 & 2 & 1 \ \alpha & 3 & -1 \ -\alpha & 1 & 2 \end{vmatrix} = 2(6 + 1) - 2(2\alpha - \alpha) + 1(\alpha + 3\alpha) = 14 - 2\alpha + 4\alpha = 14 + 2\alpha$.Substituting $\alpha = -\frac{7}{2}$:$\Delta_x = 14 + 2(-\frac{7}{2}) = 14 - 7 = 7 
eq 0$.Since $\Delta = 0$ and $\Delta_x 
eq 0$,the system is inconsistent at $\alpha = -\frac{7}{2}$.

Let the system of linear equations $x + 2y + z = 2$,$\alpha x + 3y - z = \alpha$,and $-\alpha x + y + 2z = -\alpha$ be inconsistent. Then $\alpha$ is equal to:

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