Question 6 - A-Level Maths

The matrix $\mathbf { X }$ is defined by $\left[ \begin{array} { l l } 1 & 2
3 & 0 \end{array} \right]$.
1. Given that $\mathbf { X } ^ { 2 } = \left[ \begin{array} { c c } m & 2
  3 & 6 \end{array} \right]$, find the value of $m$.
2. Show that $\mathbf { X } ^ { 3 } - 7 \mathbf { X } = n \mathbf { I }$, where $n$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
It is given that $\mathbf { A } = \left[ \begin{array} { r r } 1 & 0
0 & - 1 \end{array} \right]$.
1. Describe the geometrical transformation represented by $\mathbf { A }$.
2. The matrix $\mathbf { B }$ represents an anticlockwise rotation through $45 ^ { \circ }$ about the origin. Show that $\mathbf { B } = k \left[ \begin{array} { r r } 1 & - 1
  1 & 1 \end{array} \right]$, where $k$ is a surd.
3. Find the image of the point $P ( - 1,2 )$ under an anticlockwise rotation through $45 ^ { \circ }$ about the origin, followed by the transformation represented by $\mathbf { A }$.
  $7 \quad$ The variables $y$ and $x$ are related by an equation of the form $$y = a x ^ { n }$$ where $a$ and $n$ are constants. Let $Y = \log _ { 10 } y$ and $X = \log _ { 10 } x$.