Problem

The matrix $A$ is given by

$$A = \begin{pmatrix} 1 & -1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & a \end{pmatrix}$$

where $a \ne 1$.

(a) Find $A^{-1}$ in terms of $a$.

$$B = \begin{pmatrix} 1 & -1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{pmatrix}$$

The straight line $l_1$ is mapped onto the straight line $l_2$ by the transformation represented by the matrix $B$.

The equation of $l_2$ is

$$\bigl(\mathbf{r} - (12\mathbf{i} + 4\mathbf{j} + 6\mathbf{k})\bigr) \times (-6\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) = \mathbf{0}.$$

(b) Find a vector equation for the line $l_1$.