Matrix powers by induction

A question is this type if and only if it asks to prove a formula for M^n by mathematical induction.

5 questions · Standard +0.5

4.01a Mathematical induction: construct proofs
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CAIE Further Paper 1 2020 June Q6
13 marks Standard +0.3
6 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 1 & 1 \end{array} \right)\).
  1. The transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { - 1 }\) transforms a triangle of area \(30 \mathrm {~cm} ^ { 2 }\) into a triangle of area \(d \mathrm {~cm} ^ { 2 }\). Find the value of \(d\).
  2. Prove by mathematical induction that, for all positive integers \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 2 ^ { n } - 1 & 1 \end{array} \right)$$
  3. The line \(y = 2 x\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\), where \(\mathbf { B } = \left( \begin{array} { r l } 1 & 0 \\ 33 & 0 \end{array} \right)\). Find the value of \(n\).
OCR MEI Further Pure Core AS 2023 June Q6
8 marks Standard +0.3
6 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 } , \mathbf { M } ^ { 3 }\) and \(\mathbf { M } ^ { 4 }\).
  2. Hence make a conjecture about the matrix \(\mathbf { M } ^ { n }\).
  3. Prove your conjecture.
Edexcel CP2 2022 June Q3
11 marks Standard +0.3
  1. \(\mathbf { M } = \left( \begin{array} { l l } 3 & a \\ 0 & 1 \end{array} \right) \quad\) where \(a\) is a constant
    1. Prove by mathematical induction that, for \(n \in \mathbb { N }\)
    $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & \frac { a } { 2 } \left( 3 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$$ Triangle \(T\) has vertices \(A , B\) and \(C\).
    Triangle \(T\) is transformed to triangle \(T ^ { \prime }\) by the transformation represented by \(\mathbf { M } ^ { n }\) where \(n \in \mathbb { N }\) Given that
    • triangle \(T\) has an area of \(5 \mathrm {~cm} ^ { 2 }\)
    • triangle \(T ^ { \prime }\) has an area of \(1215 \mathrm {~cm} ^ { 2 }\)
    • vertex \(A ( 2 , - 2 )\) is transformed to vertex \(A ^ { \prime } ( 123 , - 2 )\)
    • determine
      1. the value of \(n\)
      2. the value of \(a\)
OCR FP1 AS 2021 June Q4
6 marks Standard +0.3
4 In this question you must show detailed reasoning. \(\mathbf { M }\) is the matrix \(\left( \begin{array} { l l } 1 & 6 \\ 0 & 2 \end{array} \right)\).
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right) \\ 0 & 2 ^ { n } \end{array} \right)\), for any positive integer \(n\).
OCR MEI Further Pure Core AS Specimen Q9
14 marks Challenging +1.2
You are given that matrix \(\mathbf{M} = \begin{pmatrix} -3 & 8 \\ -2 & 5 \end{pmatrix}\).
  1. Prove that, for all positive integers \(n\), \(\mathbf{M}^n = \begin{pmatrix} 1-4n & 8n \\ -2n & 1+4n \end{pmatrix}\). [6]
  2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf{M}\). [3]
It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf{M}^n\), for any positive integer \(n\).
  1. Explain geometrically why this claim is true. [2]
  2. Verify algebraically that this claim is true. [3]