Find P and D for A = PDP⁻¹

Questions asking to find matrices P and diagonal matrix D such that A = PDP⁻¹ (standard diagonalization of matrix A)

28 questions · Standard +0.7

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CAIE FP1 2019 November Q8
10 marks Challenging +1.2
The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{pmatrix},$$ where \(m \neq 0, 1, 2\).
  1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M} = \mathbf{PDP}^{-1}\). [7]
  2. Find \(\mathbf{M}^T \mathbf{P}\). [3]
CAIE FP1 2019 November Q8
10 marks Standard +0.8
The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{pmatrix},$$ where \(m \neq 0, 1, 2\).
  1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M} = \mathbf{PDP}^{-1}\). [7]
  2. Find \(\mathbf{M}^T\mathbf{P}\). [3]
AQA Further Paper 2 2019 June Q9
13 marks Challenging +1.8
  1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf{M} = \begin{bmatrix} 1 & 2 \\ 5 & 5 \\ -3 & 13 \\ 5 & 10 \end{bmatrix}$$ [5 marks]
  2. Find matrices \(\mathbf{U}\) and \(\mathbf{D}\) such that \(\mathbf{D}\) is a diagonal matrix and \(\mathbf{M} = \mathbf{U}\mathbf{D}\mathbf{U}^{-1}\) [2 marks]
  3. Given that \(\mathbf{M}^n \to \mathbf{L}\) as \(n \to \infty\), find the matrix \(\mathbf{L}\). [4 marks]
  4. The transformation represented by \(\mathbf{L}\) maps all points onto a line. Find the equation of this line. [2 marks]