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Addition, subtraction and scalar multiplication sheet
3-mark answer frame

Find the matrices X and Y if 2X + Y = [2, 2; 5, −1] and X + 2Y = [1, 4; 4, 1].

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: 2X + Y = [2, 2; 5, −1] ... (1) and X + 2Y = [1, 4; 4, 1] ... (2). To find: X and Y.
  2. 2 · Write the rule. Matrices of the same order can be added, subtracted and multiplied by scalars entry by entry, so we can eliminate one unknown as in simultaneous equations.
  3. 3 · Multiply (2) by 2. 2X + 4Y = [2, 8; 8, 2] ... (3)
  4. 4 · Subtract (1) from (3). 3Y = [0, 6; 3, 3], so Y = [0, 2; 1, 1].
  5. 5 · Find X from (2). X = [1, 4; 4, 1] − 2Y = [1, 4; 4, 1] − [0, 4; 2, 2] = [1, 0; 2, −1].
  6. 6 · Check and box the answer. 2X + Y = [2, 0; 4, −2] + [0, 2; 1, 1] = [2, 2; 5, −1], which matches (1). So X = [1, 0; 2, −1] and Y = [0, 2; 1, 1].

Scoring tip: Treat X and Y like x and y in simultaneous equations, but do every operation entry by entry.