JENECONKMaths Universe

Equality, Addition and Subtraction

Ages 13–15

Difficulty adapts to you

Equality, Addition and Subtraction

Two matrices are equal only when they have the same order and every corresponding entry is equal. To add or subtract matrices, they must have exactly the same order. Then combine entries in matching positions: (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ and (A − B)ᵢⱼ = aᵢⱼ − bᵢⱼ. Never add entries just because they are in the same-looking row if the orders differ.

Your goals

  • ✓You can decide when matrices are equal and add or subtract compatible matrices entry by entry.
  • ✓You can use corresponding entries, compatible, matrix equality correctly.
  • ✓You can compare orders before combining, then check each corresponding position.

Words to know

corresponding entries:
Entries in the same row and column position.
compatible:
Having the same order for addition or subtraction.
matrix equality:
Same order and equal corresponding entries.

Examples

  • 1A = 2514 and B = 3−162 have the same order, so A + B = 5476.
  • 2A − B = −16−52 because each entry of B is subtracted from the entry in the same position in A.
  • 3x234 = 723y gives x = 7 and y = 4 by comparing corresponding entries.
  • 4A 2 × 3 matrix cannot be added to a 3 × 2 matrix, even though both contain six entries; their layouts do not match.
  • 5Addition is commutative (A + B = B + A), but subtraction is not: A − B is usually different from B − A.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Entrywise operation

Best for: Adding or subtracting compatible matrices
  1. 1.Check that both orders are identical.
  2. 2.Circle matching positions.
  3. 3.Add or subtract each pair and keep the same order.

Example: [[2, 5], [1, 4]] + [[3, -1], [6, 2]] = [[5, 4], [7, 6]].

Method 2

Compare corresponding entries

Best for: Finding unknowns in equal matrices
  1. 1.Match entries in the same position.
  2. 2.Write one equation for each unknown.
  3. 3.Check every position before finishing.

Example: [[x, 2], [3, 4]] = [[7, 2], [3, y]] gives x = 7 and y = 4.

Check these

  • • Do not add matrices with different orders or subtract in the wrong direction.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own equality, addition and subtraction example, solve it, and explain how you checked it.

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