Practice Set-2

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Section A — Eigenvalues & Eigenvectors

2×2 Matrices

  1. Find the eigenvalues and eigenvectors of \[ A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}. \]
  2. Compute eigenvalues and eigenvectors of \[ B = \begin{bmatrix} 4 & -2 \\ 1 & 1 \end{bmatrix}. \]
  3. Find the characteristic polynomial and eigenvectors of \[ C = \begin{bmatrix} 0 & 1 \\ -4 & -5 \end{bmatrix}. \]
  4. For \[ D = \begin{bmatrix} 3 & 2 \\ 0 & 3 \end{bmatrix}, \] determine eigenvalues and check if there are enough linearly independent eigenvectors.
  5. Find eigenvalues and eigenvectors of \[ E = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}. \]

3×3 Matrices

  1. Find eigenvalues and eigenvectors of \[ F = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{bmatrix}. \]
  2. Compute the eigenvalues and eigenvectors of \[ G = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 0 & 3 \end{bmatrix}. \]
  3. Find the eigenvalues of \[ H = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 2 \end{bmatrix}, \] and determine corresponding eigenvectors.
  4. Find eigenvalues and eigenvectors of \[ I = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}. \]
  5. Compute eigenvalues and eigenvectors of \[ J = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}. \]

Section B — Cayley–Hamilton Theorem

  1. Verify Cayley–Hamilton theorem for \[ A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}. \]
  2. Use Cayley–Hamilton theorem to compute \(A^3\) for \[ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}. \]
  3. Use Cayley–Hamilton theorem to find \(A^{-1}\) for \[ A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}. \]
  4. Verify Cayley–Hamilton theorem for \[ A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & 11 & -6 \end{bmatrix}. \]
  5. Use Cayley–Hamilton theorem to compute \(A^4\) for \[ A = \begin{bmatrix} 4 & 3 \\ 3 & 2 \end{bmatrix}. \]

Section C — Diagonalization

  1. Diagonalize \[ A = \begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix}. \]
  2. Find an invertible matrix \(P\) such that \(P^{-1}AP\) is diagonal for \[ B = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}. \]
  3. Determine whether the matrix \[ C = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \] is diagonalizable. If yes, diagonalize it.
  4. Diagonalize \[ D = \begin{bmatrix} 6 & -2 \\ 2 & 3 \end{bmatrix}. \]
  5. Find the diagonal form of \[ E = \begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}. \]