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Symmetric and skew symmetric matrices sheet
3-mark answer frame

Express A = [1, 3, 5; −1, 2, 0; 3, 4, −2] as the sum of a symmetric and a skew symmetric matrix.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found. Given: A = [1, 3, 5; −1, 2, 0; 3, 4, −2]. To find: a symmetric P and a skew symmetric Q with A = P + Q.
  2. 2 · Write the formula. P = ½(A + A′) is symmetric, Q = ½(A − A′) is skew symmetric, and A = P + Q.
  3. 3 · Write A′. A′ = [1, −1, 3; 3, 2, 4; 5, 0, −2]
  4. 4 · Find P and show P′ = P.
    • A + A′ = [2, 2, 8; 2, 4, 4; 8, 4, −4]
    • P = [1, 1, 4; 1, 2, 2; 4, 2, −2]
    • Each entry equals its mirror across the diagonal, so P′ = P.
  5. 5 · Find Q and show Q′ = −Q.
    • A − A′ = [0, 4, 2; −4, 0, −4; −2, 4, 0]
    • Q = [0, 2, 1; −2, 0, −2; −1, 2, 0]
    • The diagonal is zero and each entry is the negative of its mirror, so Q′ = −Q.
  6. 6 · Check and box the answer. P + Q = [1, 3, 5; −1, 2, 0; 3, 4, −2] = A. So A = [1, 1, 4; 1, 2, 2; 4, 2, −2] + [0, 2, 1; −2, 0, −2; −1, 2, 0].

Scoring tip: Show that P′ = P and Q′ = −Q, in words or by writing the transposes. Many students lose a mark by skipping this check.