Two matrices A and B can be multiplied as AB only when the number of columns in A equals the number of rows in B. Each entry of the answer is found by multiplying a row of A by a column of B, term by term, then adding the results. Order matters: swapping the two matrices usually gives a different answer, or no answer at all, which is why AB ≠ BA in general.
Matrix multiplication looks nothing like ordinary multiplication, and most of the marks lost come from skipping the check that should happen before any calculation starts. Students jump straight into multiplying entries without confirming the two matrices are even compatible, or they multiply column by row instead of row by column, and the whole answer collapses. If you have not yet covered matrix basics like addition and scalar multiplication, start there before tackling multiplication.
The fix is to treat the compatibility check and the order of the entries as fixed rules, not optional habits. Once that discipline is in place, matrix multiplication becomes a mechanical process that is hard to get wrong. (This is the same pattern behind a lot of lost marks across the paper, as covered in why Singapore students struggle with maths.)
Step 1
Write down the order of A and B
Step 2
Check columns of A = rows of B
Step 3
Answer order is rows of A by columns of B
Step 4
Multiply each row of A by each column of B, add the products
The order of a matrix is written as rows by columns. A matrix with 2 rows and 3 columns has order 2 × 3. For A of order m × n and B of order p × q, the product AB exists only when n = p, and the answer has order m × q. Writing the two orders side by side, (m × n)(p × q), is a quick way to see this: the two middle numbers must match, and the two outer numbers give the shape of the answer.
One note on how matrices are written out below. Each matrix is listed one row at a time, so [[1, 2], [3, 4]] means the first row is 1, 2 and the second row is 3, 4. In the exam the same matrix is printed inside a single tall bracket, with the rows stacked.
Given A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], find AB.
Solution
2 × 2. Columns of A (2) equal rows of B (2), so the product exists, and the answer will be order 2 × 2.(1 × 5) + (2 × 7) = 5 + 14 = 19.(1 × 6) + (2 × 8) = 6 + 16 = 22.(3 × 5) + (4 × 7) = 15 + 28 = 43.(3 × 6) + (4 × 8) = 18 + 32 = 50.AB = [[19, 22], [43, 50]].Quick check: before writing any number, say the compatibility check out loud: "columns of A equal rows of B." If a question gives you two matrices where this fails, the question is almost certainly testing whether you spot that the product does not exist, not asking you to force an answer.
Given A = [[2, 0, 1], [1, 3, 2]], order 2 × 3, and B = [[1, 2], [0, 1], [3, 1]], order 3 × 2, find AB.
Solution
2 × 2.(2, 0, 1), by column 1 of B, (1, 0, 3). (2 × 1) + (0 × 0) + (1 × 3) = 2 + 0 + 3 = 5.(2, 1, 1). (2 × 2) + (0 × 1) + (1 × 1) = 4 + 0 + 1 = 5.(1, 3, 2), by column 1 of B. (1 × 1) + (3 × 0) + (2 × 3) = 1 + 0 + 6 = 7.(1 × 2) + (3 × 1) + (2 × 1) = 2 + 3 + 2 = 7.AB = [[5, 5], [7, 7]].The step students get wrong
Multiplying entry by entry instead of row by column, the same mistake as adding two matrices, is the single most common error here. In matrix addition, you match position to position. In matrix multiplication, each answer entry needs an entire row from the first matrix paired with an entire column from the second, then those pairs are multiplied and summed. Treat the two operations as unrelated procedures, and check after every entry that you used a full row and a full column, not a single position.
Matrix multiplication is not commutative, meaning the order you multiply in changes the answer, or whether an answer exists at all. Take the two matrices from worked example 2: A is order 2 × 3 and B is order 3 × 2. The product AB exists, as shown above. But for BA, check the compatibility rule again: columns of B (2) must equal rows of A (2). That does hold here, so BA also exists, but as a 3 × 3 matrix, a completely different shape from the 2 × 2 answer for AB. Different order means the two products cannot even be compared, let alone be equal.
Even when both A and B are square and both AB and BA exist with the same order, the entries are usually still different. This is the key idea examiners test: never assume AB = BA unless the question specifically proves it for that pair of matrices.
Example: same order, different answer
Let A = [[1, 1], [0, 1]] and B = [[2, 0], [1, 1]].
AB: top-left (1×2)+(1×1)=3, top-right (1×0)+(1×1)=1, bottom-left (0×2)+(1×1)=1, bottom-right (0×0)+(1×1)=1. So AB = [[3, 1], [1, 1]].BA: top-left (2×1)+(0×0)=2, top-right (2×1)+(0×1)=2, bottom-left (1×1)+(1×0)=1, bottom-right (1×1)+(1×1)=2. So BA = [[2, 2], [1, 2]].AB ≠ BA, confirming that swapping the order changes the result even for two square matrices of the same size.Exam tip: when a question asks you to find both AB and BA, do not shortcut by assuming they match. Compute both fully. Some questions deliberately choose matrices where one product exists and the other does not, so always redo the compatibility check for the second product separately.
How do I remember the compatibility rule?
Write the two orders next to each other, for example (2 × 3)(3 × 2). The two numbers in the middle must be equal for the product to exist, and the two numbers on the outside give you the order of the answer, in this case 2 × 2.
Can I multiply two matrices of the same order, like two 2 × 2 matrices?
Yes, since the columns of the first will always equal the rows of the second when both matrices are square and the same size. This is why square matrices of matching order can always be multiplied in either direction, though the two products will usually still differ.
What does it mean if a question says a product does not exist?
It means the compatibility rule fails for that pairing, the columns of the first matrix do not equal the rows of the second. State this clearly in your answer rather than attempting a calculation, since forcing numbers together when the sizes do not match will not earn method marks.
Is matrix multiplication used anywhere outside pure calculation questions?
In the O-Level syllabus, matrices also appear in short data-handling questions, where a row or column matrix of quantities is multiplied by a matrix of prices or weights to get a total. The same row-by-column method applies, just with a smaller matrix on one side. Matrices sit in the E-Maths syllabus, so if you are weighing up both papers, our comparison of E-Maths and A-Maths shows how the two differ.
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Mr. Gan works with students who want a systematic method for matrices that holds up under exam pressure, not a rule half-remembered from class.
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