(a) The matrix \(\mathbf{M}\) can be decomposed into two transformations: a one-way stretch with scale factor 3 parallel to the x-axis, followed by a rotation anticlockwise about the origin through an angle \(\theta\).

(b) The matrix \(\mathbf{M} = \begin{pmatrix} 3\cos \theta & -\sin \theta \\ 3\sin \theta & \cos \theta \end{pmatrix}\) transforms \(\begin{pmatrix} x \\ y \end{pmatrix}\) to \(\begin{pmatrix} 3x\cos \theta - y\sin \theta \\ 3x\sin \theta + y\cos \theta \end{pmatrix}\).

For an invariant line through the origin, \(y = mx\) transforms to \(Y = mX\), leading to the equation:

\(3x\sin \theta + mx\cos \theta = 3mx\cos \theta - m^2x\sin \theta\)

\(m^2 \sin \theta - 2m \cos \theta + 3 \sin \theta = 0\)

Setting the discriminant equal to zero:

\(4\cos^2 \theta - 12\sin^2 \theta = 0\)

Using \(\tan \theta = \pm \frac{1}{\sqrt{3}}\) or \(4\cos^2 \theta - 12(1 - \cos^2 \theta) = 0\) leads to \(16\cos^2 \theta - 12 = 0\) leading to \(\cos \theta = \pm \frac{\sqrt{3}}{2}\).

Thus, \(\theta = \frac{1}{6} \pi\) and \(\theta = \frac{5}{6} \pi\).