Matrices And Determinants

Let S\(_1\) and S\(_2\) be respectively the sets of all \(a \in \mathbb{R} - \{ 0\}\) for which the system of linear equations

\(ax + 2ay - 3az = 1\)

\((2a + 1)x + (2a + 3)y + (a + 1)z = 2\)

\((3a + 5)x + (a + 5)y + (a + 2)z = 3\)

has unique solution and infinitely many solutions. Then