Diagonalisation and Powers of a Matrix · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Eigenbases make powers easy

Mathematics · Linear Algebra · ages 19-20
Name ______________________   Date ____________

  1. A has eigenvectors (1, 0) and (1, 1), with matching eigenvalues 4 and 6. What is the top-left entry of A?

    Answer: ______________

  2. What is the largest eigenvalue of the diagonal matrix with diagonal entries 4 and 7?

    Answer: ______________

  3. A real 2 by 2 diagonalisation has two different entries on D's diagonal. Lee swaps P's columns but leaves D unchanged. Lee claims this still diagonalises the same A. Is Lee right?

    Circle one:   True   False

  4. In A = P D P inverse, what sits on the diagonal of D?

    • The eigenvalues of A
    • The eigenvectors of A
    • The diagonal entries of P

  5. For A = 3I, P = [[1, 2], [0, 0]] and D = diag(3, 3) give a valid diagonalisation A = P D P inverse.

    Circle one:   True   False

  6. A = P diag(2, 4) P inverse, with P = [[1, 1], [0, 1]]. What is the top-right entry of A cubed?

    Answer: ______________

  7. Which matrix cannot be diagonalised?

    • [[1, 0], [0, 3]]
    • [[2, 0], [0, 2]]
    • [[2, 1], [0, 2]]

  8. A = P D P inverse with D = diag(2, 3). What is the trace of A squared?

    Answer: ______________

  9. A 2 by 2 matrix has eigenvalues 1 and 2. What is the trace of A to the 10th?

    Answer: ______________

  10. In an eigenvector basis, A = diag(1, 0.5). A vector starts with coordinates (3, 32). After five applications of A, what is its second coordinate?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Eigenbases make powers easy W1-mt_w15uCVhdhX-s1

  1. 4 · A(1, 0) is A's first column. It equals 4(1, 0), so the top-left entry is 4.
  2. 7 · The diagonal entries 4 and 7 are the eigenvalues. The largest is 7.
  3. False · Swapping P's columns requires swapping the matching entries of D. With different eigenvalues, leaving D unchanged mismatches the eigenpairs.
  4. The eigenvalues of A · D holds the eigenvalues in the order matching P's eigenvector columns.
  5. False · Although both columns are eigenvectors, they are dependent. P has no inverse.
  6. 56 · For this P, the top-right entry of A cubed is 4 cubed minus 2 cubed = 64 - 8 = 56.
  7. [[2, 1], [0, 2]] · [[2, 1], [0, 2]] has eigenvalue 2 twice but only one eigenvector direction, so no eigenbasis exists.
  8. 13 · A squared has eigenvalues 4 and 9, and trace = 4 + 9 = 13.
  9. 1025 · The eigenvalues raised to the 10th power are 1 and 1024. Their sum gives trace = 1025.
  10. 1 · The second coordinate is multiplied by 0.5 five times: 32 times (0.5 to the fifth) = 1.
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