Problem

(i) The matrix $\mathbf{A}$ is defined by

$\mathbf{A} = \begin{pmatrix} 3k & 4k - 1 \\ 2 & 6 \end{pmatrix}$

where $k$ is a constant.

(a) Determine the value of $k$ for which $\mathbf{A}$ is singular.

Given that $\mathbf{A}$ is non-singular,

(b) determine $\mathbf{A}^{-1}$ in terms of $k$, giving your answer in simplest form.

(ii) The matrix $\mathbf{B}$ is defined by

$\mathbf{B} = \begin{pmatrix} p & 0 \\ 0 & q \end{pmatrix}$

where $p$ and $q$ are integers.

State the value of $p$ and the value of $q$ when $\mathbf{B}$ represents

(a) an enlargement about the origin with scale factor $-2$

(b) a reflection in the $y$-axis.