(Solved): Consider the system of equations \[ \begin{array}{rrrrrr} x+ & 2 y+ & z & = & ...
Consider the system of equations \[ \begin{array}{rrrrrr} x+ & 2 y+ & z & = & -k \\ x+ & (2+k) y+ & (1-k) z & = & 0 \\ x+ & 2 y+ & \left(1-k+k^{2}\right) z & = & -1 \end{array} \] where \( x, y, z \in \mathbb{R} \) and \( k \in \mathbb{R} \). Determine the values of \( k \), if any, for which the system has (i) a unique solution, (ii) no solutions, (iii) infinitely many solutions. Find all solutions to the system when \( k=1 \).
Expert Answer
The system of equations where x,y,z?R and k?R are as follows: x+2y+z=?kx+(2+k)y+(1?k)z=0x+2y+(1?k+k2)z=?1 In matrix form, ?[12112+k1?k121?k+k2][xyz]=[