The cubic equation \(x^3 + bx^2 + cx - 1 = 0\), where \(b\) and \(c\) are constants, has roots \(\alpha, \beta, \gamma\).

It is given that the matrix \(\begin{pmatrix} 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{pmatrix}\) is singular.

(a) Show that \(\alpha^2 + \beta^2 + \gamma^2 = 3\).

(b) It is given that \(\alpha^3 + \beta^3 + \gamma^3 = 3\) and that the constants \(b\) and \(c\) are positive.

Find the values of \(b\) and \(c\).