Prove matrix identity or property

A question is this type if and only if it asks to prove an algebraic result about matrices such as M^n = kI, (A+B)^2 = kI, or verify a matrix equation.

2 questions · Standard +0.8

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OCR FP1 2005 June Q9
12 marks Standard +0.3
9
  1. Write down the matrix \(\mathbf { C }\) which represents a stretch, scale factor 2 , in the \(x\)-direction.
  2. The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { D }\).
  3. The matrix \(\mathbf { M }\) represents the combined effect of the transformation represented by \(\mathbf { C }\) followed by the transformation represented by \(\mathbf { D }\). Show that $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 1 \end{array} \right)$$
  4. Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)\), for all positive integers \(n\).
AQA Further Paper 2 2022 June Q13
16 marks Challenging +1.2
13
  1. The matrix A represents a reflection in the line \(y = m x\), where \(m\) is a constant. Show that \(\mathbf { A } = \left( \frac { 1 } { m ^ { 2 } + 1 } \right) \left[ \begin{array} { c c } 1 - m ^ { 2 } & 2 m \\ 2 m & m ^ { 2 } - 1 \end{array} \right]\) You may use the result in the formulae booklet. 13
  2. \(\quad\) The matrix \(\mathbf { B }\) is defined as \(\mathbf { B } = \left[ \begin{array} { l l } 3 & 0 \\ 0 & 3 \end{array} \right]\) Show that \(( \mathbf { B A } ) ^ { 2 } = k \mathbf { I }\) where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix and \(k\) is an integer.
    13
    1. The diagram below shows a point \(P\) and the line \(y = m x\) Draw four lines on the diagram to demonstrate the result proved in part (b).
      Label as \(P ^ { \prime }\) the image of \(P\) under the transformation represented by (BA) \({ } ^ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-20_579_1068_584_488} 13
  4. (ii) Explain how your completed diagram shows the result proved in part (b).
    13
  5. The matrix \(\mathbf { C }\) is defined as \(\mathbf { C } = \left[ \begin{array} { c c } \frac { 12 } { 5 } & \frac { 9 } { 5 } \\ \frac { 9 } { 5 } & - \frac { 12 } { 5 } \end{array} \right]\) Find the value of \(m\) such that \(\mathbf { C } = \mathbf { B A }\) Fully justify your answer.
    [0pt] [4 marks]