Matrices
How to Add and subtract matrices
Use this to add or subtract two matrices of the same order.
Before you start
- Knowing the order of a matrix (rows × columns)
- Adding and subtracting negative numbers
- Matching entries by position
When to use it
Use this to add or subtract two matrices of the same order.
The steps
- Check both matrices have the same order (same rows and columns).
- Add or subtract the entries in matching positions.
- Keep each result in its original position.
- Write the answer as a matrix of the same order.
- Double-check the signs when subtracting.
Worked example
Given A = (3 1; 2 4) and B = (5 0; 1 2), find A + B.
- Both matrices are 2×2, so they have the same order and can be added.
- Add the entries in matching positions: 3 + 5, 1 + 0, 2 + 1, 4 + 2.
- Keep each result in its own position: 8 top-left, 1 top-right, 3 bottom-left, 6 bottom-right.
- Write the answer as a 2×2 matrix: A + B = (8 1; 3 6).
- A quick check of the additions confirms every entry.
A second example, with a twist
This is a subtraction with negative entries, where a step like 0 − (−6) turns into an addition, so watch the signs. Given A = (4 −2; 0 5) and B = (1 3; −6 5), find A − B.
- Both matrices are 2×2, so they have the same order and can be subtracted.
- Subtract the entries in matching positions: 4 − 1, −2 − 3, 0 − (−6), 5 − 5.
- Keep each result in its own position: 3 top-left, −5 top-right, 0 − (−6) = 6 bottom-left, 0 bottom-right.
- Write the answer as a 2×2 matrix: A − B = (3 −5; 6 0).
- Double-check the signs when subtracting, especially 0 − (−6) = 6 and −2 − 3 = −5.
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
Can I add matrices of different sizes?
No. To add or subtract, both matrices must have exactly the same order, the same number of rows and the same number of columns.
A 2×2 and a 2×3 cannot be added, because some positions would have no partner to combine with. Always check the order first.
Does the position of each entry matter?
Yes, very much. You only combine entries that sit in the same position, top-left with top-left, and so on, and the result stays in that same position.
Mixing up positions gives a wrong matrix even if every individual sum is correct. Line the two matrices up carefully.
What is the most common mistake when subtracting?
Getting the signs wrong. Subtracting a negative entry becomes an addition, so 0 − (−6) is 6, not −6.
And a positive minus a bigger positive gives a negative, like −2 − 3 = −5. Take each subtraction slowly and keep every minus sign attached to its number.