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Determinantshard
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Consider the system of linear equations in x,y,zx, y, zx,y,z with a parameter aaa: {ax+y+z=1x+ay+z=ax+y+az=a2\begin{cases} a x + y + z = 1 \\ x + a y + z = a \\ x + y + a z = a^2 \end{cases}⎩⎨⎧​ax+y+z=1x+ay+z=ax+y+az=a2​ For which values of aaa does the system have a unique solution, and what is the value of zzz when a=3a = 3a=3?