The linear transformation \(\mathrm{T} : \mathbb{R}^4 \to \mathbb{R}^4\) is represented by the matrix \(\mathbf{M}\), where
$$\mathbf{M} = \begin{pmatrix} 3 & 2 & 0 & 1 \\ 6 & 5 & -1 & 3 \\ 9 & 8 & -2 & 5 \\ -3 & -2 & 0 & -1 \end{pmatrix}.$$
- Find the rank of \(\mathbf{M}\). [3]
Let \(K\) be the null space of \(\mathrm{T}\).
- Find a basis for \(K\). [3]
- Find the general solution of
$$\mathbf{M}\mathbf{x} = \begin{pmatrix} 2 \\ 5 \\ 8 \\ -2 \end{pmatrix}.$$ [3]