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With ordinary numbers, 3 × 5 and 5 × 3 are both 15, so the order never matters. Matrices break that habit. The ICSE Class 10 syllabus for 2026-27 includes "multiplication of a 2×2 matrix by a matrix" under the Matrices unit (syllabus as reproduced by Oswal Publishers), and multiplying in the wrong order is an easy way to lose marks there.
What happens when you multiply two matrices both ways?
We write each 2×2 matrix row by row, with a semicolon between rows: A = [1 2; 3 4] and B = [0 1; 1 0].
To find AB, pair each row of A with each column of B:
Row 1 of A (1, 2) with column 1 of B (0, 1): 1×0 + 2×1 = 2
Row 1 of A (1, 2) with column 2 of B (1, 0): 1×1 + 2×0 = 1
Row 2 of A (3, 4) with column 1 of B: 3×0 + 4×1 = 4
Row 2 of A (3, 4) with column 2 of B: 3×1 + 4×0 = 3
So AB = [2 1; 4 3].
For BA, the rows come from B and the columns from A:
Row 1 of B (0, 1) with column 1 of A (1, 3): 0×1 + 1×3 = 3
Row 1 of B (0, 1) with column 2 of A (2, 4): 0×2 + 1×4 = 4
Row 2 of B (1, 0) with column 1 of A: 1×1 + 0×3 = 1
Row 2 of B (1, 0) with column 2 of A: 1×2 + 0×4 = 2
So BA = [3 4; 1 2]. Same two matrices, two different answers.
Why is AB not equal to BA?
Every entry of a product is built from a row of the left matrix and a column of the right matrix. Swap the order and you swap which numbers get paired, so the entries change.
AB = [2 1; 4 3] is A with its columns swapped. BA = [3 4; 1 2] is A with its rows swapped. B is a "swap" matrix, and whether it swaps rows or columns depends only on which side of A it sits.
An operation is commutative when AB is defined, BA is also defined, and AB = BA (Matrix multiplication, Wikipedia). The same article explains that BA stands for doing A first, then B, and two moves done in reverse order generally give a different result.
Sometimes BA does not even exist. The ICSE 2024 specimen paper, Question 1(i), gave A as a 1×2 matrix and B as a 2×2 matrix, with both AB and BA among the options; the answer is AB (solved specimen 2024, KnowledgeBoat). BA fails because B has two columns but A has only one row. Our post on when two matrices can be multiplied covers that size check.
Where does the AB ≠ BA trap show up in ICSE papers?
In the ICSE 2025 Class 10 maths paper, Question 3(ii) gave A = [1 2; 3 4], B = [2 1; 4 2] and C = [-5 1; 7 -4], and asked for B(A + C) − 5B in steps, starting with A + C (solved 2025 paper, KnowledgeBoat).
A + C = [-4 3; 10 0]. With B on the left, B(A + C) = [2 6; 4 12], and the final answer is B(A + C) − 5B = [-8 1; -16 2].
Now suppose you write (A + C)B instead. You get [4 2; 20 10], and the last part becomes [-6 -3; 0 0]. Each multiplication is right; the order is wrong.
The same habit breaks familiar algebra: (A + B)² = A² + AB + BA + B², which equals A² + 2AB + B² only if AB = BA. We work through that in why (A + B)² fails for matrices. For where matrices sit in the whole paper, see our guide to the ICSE Class 10 maths paper units.
How do you avoid the order mistake in the exam?
Use this four-point check on every matrix product:
Copy the product exactly as printed. B(A + C) means B goes on the left.
Write the order under each matrix, such as (2×2)(2×2), and say it as you work: rows of the left, columns of the right.
When you open brackets, keep the sides: B(A + C) = BA + BC, never AB + CB.
Never write 2AB for AB + BA unless the question tells you the matrices commute.
If you keep getting the method right but the order wrong, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. The Fermi Concept Library has its own lesson for each mathematics concept.
Next, do the self-test above, then redo ICSE 2025 Question 3(ii) with B on the left.
