Add matrices by adding entries in the same position, and multiply them by combining each row of the first with each column of the second. Adding needs matching orders, and multiplying needs the first matrix’s columns to equal the second matrix’s rows.
This lesson follows reading matrix order and entries in the SPM Mathematics matrices chapter.
How do you add matrices?
Check that both orders are equal, then add matching entries. There is no other rule.
Take A and B, both of order 2 × 2:
A = | 2 3 | B = | 5 0 |
| 1 4 | | −2 1 |A + B has entries 2 + 5 = 7, 3 + 0 = 3, 1 + (−2) = −1 and 4 + 1 = 5. So A + B has rows (7, 3) and (−1, 5).
How do you multiply matrices?
Each entry of the product comes from one row of the first matrix and one column of the second. Multiply along, then add.
For AB, the entry in row 1, column 1 uses row 1 of A (2, 3) and column 1 of B (5, −2):
2 × 5 + 3 × (−2) = 10 − 6 = 4.
The four entries of AB are:
| Position | Working | Value |
|---|---|---|
| Row 1, column 1 | 2(5) + 3(−2) | 4 |
| Row 1, column 2 | 2(0) + 3(1) | 3 |
| Row 2, column 1 | 1(5) + 4(−2) | −3 |
| Row 2, column 2 | 1(0) + 4(1) | 4 |
So AB has rows (4, 3) and (−3, 4).
Why does the order matter?
Now work out BA with the same matrices. Row 1 of B is (5, 0), and row 2 is (−2, 1).
BA has rows (5(2) + 0(1), 5(3) + 0(4)) = (10, 15), and (−2(2) + 1(1), −2(3) + 1(4)) = (−3, −2).
AB has rows (4, 3) and (−3, 4), but BA has rows (10, 15) and (−3, −2). They are different matrices, so the order in the question must be kept.
The mistake that costs marks
The common slip is to multiply entry by entry, as if it were addition. That gives 2 × 5 = 10, 3 × 0 = 0, 1 × (−2) = −2 and 4 × 1 = 4, which is the matrix with rows (10, 0) and (−2, 4).
| Wrong | Right | |
|---|---|---|
| Method | Same position times same position | Row times column, then add |
| Row 1, column 1 | 10 | 4 |
| Check | Skipped the order test | Columns of A = rows of B |
The answer looks tidy, which makes the error hard to see. Before any product, write the two orders and check that the inner numbers match.
Check yourself
Given C with rows (1, 2) and (3, 0), and D with rows (4, 1) and (2, 5), find CD.
Answer
Both matrices are 2 × 2, so CD exists and has order 2 × 2.
Row 1: 1(4) + 2(2) = 8, and 1(1) + 2(5) = 11.
Row 2: 3(4) + 0(2) = 12, and 3(1) + 0(5) = 3.
CD has rows (8, 11) and (12, 3).
What to study next
The inverse is defined through multiplication, so continue with finding the inverse of a two-by-two matrix. You can test your own pairs in the two-by-two matrix operations tutor, and record slips in the mistake log and paper-error review.
If you want a teacher to check your products as you write them, see online one-to-one Mathematics tuition.