Given Information
We are given with following system of equations: \[ \begin{aligned} s x _ { 1 } - 2 x _ { 2 } & = 1 \\ 4 s x _ { 1 } + 4 s x _ { 2 } & = 2 \end{aligned} \] We have to find the values of the parameter s for which the system has a unique solution, and describe the solution.

Step-1: The matrices
The matrices A, and B for the equation $Ax=B$ are: \[ A = \left[ \begin{array} { c c } { s } & { - 2 } \\ { 4 s } & { 4 s } \end{array} \right] \text { and } \mathbf { b } = \left[ \begin{array} { l } { 1 } \\ { 2 } \end{array} \right] \] The matrices $ A _ { 1 } ( \mathbf { b } ) \text { and } A _ { 2 } ( \mathbf { b } ) $ are: \[ A _ { 1 } ( \mathbf { b } ) = \left[ \begin{array} { c c } { 1 } & { - 2 } \\ { 2 } & { 4 s } \end{array} \right] A _ { 2 } ( \mathbf { b } ) = \left[ \begin{array} { c c } { s } & { 1 } \\ { 4 s } & { 2 } \end{array} \right] \]

Step-2: The Determinants
Determinant of A: \[ \begin{aligned} \operatorname { det } A & = ( s ) ( 4 s ) - ( - 2 ) ( 4 s ) \\ & = 4 s ^ { 2 } + 8 s \\ & = 4 s ( s + 2 ) \end{aligned} \] Determinant of [$A_1(b)$]: \[ \begin{aligned} \operatorname { det } \left[ A _ { 1 } ( \mathbf { b } ) \right] & = ( 4 s ) ( 1 ) - ( - 2 ) ( 2 ) \\ & = 4 s + 4 \end{aligned} \] Determinant of [$A_2(b)$]: \[ \begin{aligned} \operatorname { det } \left[ A _ { 2 } ( \mathbf { b } ) \right] & = ( s ) ( 2 ) - ( 4 s ) ( 1 ) \\ & = 2 s - 4 s \\ & = - 2 s \end{aligned} \]

Step-3: The parameter
For, unique solution, the determinant of A should be nonzero \[ \begin{aligned} 4 s ( s + 2 ) & = 0 \\ s & = - 2,0 \end{aligned} \] Therefore, the system has unique solution if

\[ s \neq -2 , s \neq 0 \]

Step-4: The Solution
Solution of $x_1$ \[ \begin{aligned} x _ { 1 } & = \dfrac { \operatorname { det } \left[ A _ { 1 } ( \mathbf { b } ) \right] } { \operatorname { det } A } \\ & = \dfrac { 4 s + 4 } { 4 s ( s + 2 ) } \\ & = \dfrac { s + 1 } { s ( s + 2 ) } \end{aligned} \] Solution of $x_2$ \[ \begin{aligned} x _ { 2 } & = \dfrac { \operatorname { det } \left[ A _ { 2 } ( \mathbf { b } ) \right] } { \operatorname { det } A } \\ & = \dfrac { - 2 s } { 4 s ( s + 2 ) } \\ & = \dfrac { - 1 } { 2 ( s + 2 ) } \end{aligned} \] Final Solution:

\[ \left[ \begin{array} { c } { \dfrac { s + 1 } { s ( s + 2 ) } } \\ { \dfrac { - 1 } { 2 ( s + 2 ) } } \end{array} \right] \]