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(B) The given system of equations is:$x+y+z=1$$2x+3y+2z=2$$ax+ay+2az=4$$A$ system of linear equations $AX=B$ has a unique solution if and only if the

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For all $a \in R-\{0\}$

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(B) The given system of equations is:$x+y+z=1$$2x+3y+2z=2$$ax+ay+2az=4$$A$ system of linear equations $AX=B$ has a unique solution if and only if the determinant of the coefficient matrix $A$ is non-zero,i.e.,$|A| 
eq 0$.The coefficient matrix $A$ is:$A = \begin{bmatrix} 1 & 1 & 1 \ 2 & 3 & 2 \ a & a & 2a \end{bmatrix}$Calculating the determinant $|A|$:$|A| = 1(3(2a) - 2(a)) - 1(2(2a) - 2(a)) + 1(2(a) - 3(a))$$|A| = 1(6a - 2a) - 1(4a - 2a) + 1(2a - 3a)$$|A| = 4a - 2a - a = a$For a unique solution,we require $|A| 
eq 0$,which implies $a 
eq 0$.Therefore,the system has a unique solution for all $a \in R-\{0\}$.

For what values of $a$ will the system of equations $x+y+z=1$,$2x+3y+2z=2$,and $ax+ay+2az=4$ have a unique solution?

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