Matrix powers and repeated transformations

Edexcel F1 2022 June Q7

9 marks Standard +0.3

7. $$A = \left( \begin{array} { c c } - \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

Determine the matrix $\mathbf { A } ^ { 2 }$
Describe fully the single geometrical transformation represented by the matrix $\mathbf { A } ^ { 2 }$
Hence determine the smallest positive integer value of $n$ for which $\mathbf { A } ^ { n } = \mathbf { I }$ The matrix $\mathbf { B }$ represents a stretch scale factor 4 parallel to the $x$-axis.
Write down the matrix $\mathbf { B }$ The transformation represented by matrix $\mathbf { A }$ followed by the transformation represented by matrix $\mathbf { B }$ is represented by the matrix $\mathbf { C }$
Determine the matrix $\mathbf { C }$ The parallelogram $P$ is transformed onto the parallelogram $P ^ { \prime }$ by the matrix $\mathbf { C }$
Given that the area of parallelogram $P ^ { \prime }$ is 20 square units, determine the area of parallelogram $P$

OCR FP1 2010 January Q10

11 marks Standard +0.8

10 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)$.

Find $\mathbf { M } ^ { 2 }$ and $\mathbf { M } ^ { 3 }$.
Hence suggest a suitable form for the matrix $\mathbf { M } ^ { n }$.
Use induction to prove that your answer to part (ii) is correct.
Describe fully the single geometrical transformation represented by $\mathbf { M } ^ { 10 }$.

OCR MEI FP1 2012 January Q9

12 marks Standard +0.3

9 The matrix $\mathbf { R }$ is $\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)$.

Explain in terms of transformations why $\mathbf { R } ^ { 4 } = \mathbf { I }$.
Describe the transformation represented by $\mathbf { R } ^ { - 1 }$ and write down the matrix $\mathbf { R } ^ { - 1 }$.
$\mathbf { S }$ is the matrix representing rotation through $60 ^ { \circ }$ anticlockwise about the origin. Find $\mathbf { S }$.
Write down the smallest positive integers $m$ and $n$ such that $\mathbf { S } ^ { m } = \mathbf { R } ^ { n }$, explaining your answer in terms of transformations.
Find $\mathbf { R S }$ and explain in terms of transformations why $\mathbf { R S } = \mathbf { S R }$. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

OCR MEI Further Pure Core 2024 June Q8

10 marks Standard +0.3

8

Specify fully the transformation T of the plane associated with the matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { l l } 1 & \lambda \\ 0 & 1 \end{array} \right)$ and $\lambda$ is a non-zero constant.
1. Find detM.
2. Deduce two properties of the transformation T from the value of detM.
Prove that $\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & n \lambda \\ 0 & 1 \end{array} \right)$, where $n$ is a positive integer.
Hence specify fully a single transformation which is equivalent to $n$ applications of the transformation T.

OCR Further Pure Core AS 2023 June Q9

10 marks Standard +0.8

9 Matrix $\mathbf { R }$ is given by $\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b \\ 0 & 1 & 0 \\ b & 0 & a \end{array} \right)$ where $a$ and $b$ are constants.

Find $\mathbf { R } ^ { 2 }$ in terms of $a$ and $b$. The constants $a$ and $b$ are given by $a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )$ and $b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )$.
By determining exact expressions for $a b$ and $a ^ { 2 } - b ^ { 2 }$ and using the result from part (a), show that $\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1 \\ 0 & 2 & 0 \\ 1 & 0 & \sqrt { 3 } \end{array} \right)$ where $k$ is a real number whose value is to be determined.
Find $\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }$ and $\mathbf { R } ^ { 24 }$.
Describe fully the transformation represented by $\mathbf { R }$. \section*{END OF QUESTION PAPER}