Describe single transformation from matrix

A question is this type if and only if it asks to describe fully the geometrical transformation represented by a single given matrix (e.g. rotation, reflection, enlargement, stretch, shear).

33 questions · Moderate -0.5

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OCR MEI Further Pure Core AS 2023 June Q1
3 marks Easy -1.2
1 The transformation R of the plane is reflection in the line \(x = 0\).
  1. Write down the matrix \(\mathbf { M }\) associated with R .
  2. Find \(\mathbf { M } ^ { 2 }\).
  3. Interpret the result of part (b) in terms of the transformation \(R\).
OCR MEI Further Pure Core AS 2020 November Q4
4 marks Moderate -0.3
4 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r r } 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)\).
    1. Calculate \(\operatorname { det } \mathbf { M }\).
    2. State two geometrical consequences of this value for the transformation associated with \(\mathbf { M }\).
  2. Describe fully the transformation associated with \(\mathbf { M }\).
OCR MEI Further Pure Core AS 2020 November Q8
7 marks Moderate -0.3
8
  1. The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\).
    1. Find \(\mathbf { M } ^ { 2 }\).
    2. Write down the transformation represented by \(\mathbf { M }\).
    3. Hence state the geometrical significance of the result of part (i).
  2. The matrix \(\mathbf { N }\) is \(\left( \begin{array} { c c } k + 1 & 0 \\ k & k + 2 \end{array} \right)\), where \(k\) is a constant. Using determinants, investigate whether \(\mathbf { N }\) can represent a reflection.
OCR MEI Further Pure Core 2019 June Q11
12 marks Standard +0.3
11
  1. Specify fully the transformations represented by the following matrices.
    • \(\mathbf { M } _ { 1 } = \left( \begin{array} { r r } \frac { 3 } { 5 } & - \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\)
    • \(\mathbf { M } _ { 2 } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\)
    • Find the equation of the mirror line of the reflection R represented by the matrix \(\mathbf { M } _ { 3 } = \mathbf { M } _ { 1 } \mathbf { M } _ { 2 }\).
    • It is claimed that the reflection represented by the matrix \(\mathbf { M } _ { 4 } = \mathbf { M } _ { 2 } \mathbf { M } _ { 1 }\) has the same mirror line as R . Explain whether or not this claim is correct.
Edexcel CP AS 2021 June Q1
7 marks Easy -1.2
1. $$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$
    1. Describe fully the single geometrical transformation \(P\) represented by the matrix \(\mathbf { P }\).
    2. Describe fully the single geometrical transformation \(Q\) represented by the matrix \(\mathbf { Q }\). The transformation \(P\) followed by the transformation \(Q\) is the transformation \(R\), which is represented by the matrix \(\mathbf { R }\).
  2. Determine \(\mathbf { R }\).
    1. Evaluate the determinant of \(\mathbf { R }\).
    2. Explain how the value obtained in (c)(i) relates to the transformation \(R\).
OCR FP1 2011 January Q7
9 marks Moderate -0.8
  1. Write down the matrix, \(\mathbf { A }\), that represents a shear with \(x\)-axis invariant in which the image of the point \(( 1,1 )\) is \(( 4,1 )\).
  2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0 \\ 0 & \sqrt { 3 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
  3. The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 2 & 6 \\ 0 & 2 \end{array} \right)\).
    (a) Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
    (b) Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\). 8 The quadratic equation \(2 x ^ { 2 } - x + 3 = 0\) has roots \(\alpha\) and \(\beta\), and the quadratic equation \(x ^ { 2 } - p x + q = 0\) has roots \(\alpha + \frac { 1 } { \alpha }\) and \(\beta + \frac { 1 } { \beta }\).
  4. Show that \(p = \frac { 5 } { 6 }\).
  5. Find the value of \(q\). 9 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1 \\ 3 & a & 1 \\ 4 & 2 & 1 \end{array} \right)\).
  6. Find, in terms of \(a\), the determinant of \(\mathbf { M }\).
  7. Hence find the values of \(a\) for which \(\mathbf { M } ^ { - 1 }\) does not exist.
  8. Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k \\ & 3 x + 6 y + z = 0 \\ & 4 x + 2 y + z = k \end{aligned}$$ where \(k\) is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
  9. Show that \(\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }\).
  10. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
  11. Show that \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }\).
AQA FP1 2006 June Q5
9 marks Moderate -0.3
5 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$
  1. Find the matrix:
    1. \(\mathbf { M } ^ { 2 }\);
    2. \(\mathbf { M } ^ { 4 }\).
  2. Describe fully the geometrical transformation represented by \(\mathbf { M }\).
  3. Find the matrix \(\mathbf { M } ^ { 2006 }\).
AQA Further AS Paper 1 Specimen Q1
1 marks Easy -1.8
1 A reflection is represented by the matrix \(\left[ \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right]\) State the equation of the line of invariant points. Circle your answer.
[0pt] [1 mark] $$x = 0 \quad y = 0 \quad y = x \quad y = - x$$