Rows of A, columns of B
For A = [1, 2; 0, 1] and B = [2, 0; 1, 3], find AB.
- Top left: 1×2 + 2×1 = 4; top right: 1×0 + 2×3 = 6.
- Bottom left: 0×2 + 1×1 = 1; bottom right: 0×0 + 1×3 = 3.
Answer: [4, 6; 1, 3]
Matrix multiplication
Calculate AB and BA for two matrices and compare the results.
Before you start
You’ll need Expanding and factorising. Follow a link if you want to revise it first.
Row by column
Each entry of AB is the dot product of one row of A and one column of B. Multiply corresponding entries, then add.
AB and BA usually differ. Matrix multiplication is associative when the products exist, but not generally commutative.
When matrices act on column vectors, AB applies B first and then A. This mirrors the inner-first order of function composition.
The examples give AB ≠ BA. This single counterexample disproves the claim that matrix multiplication is always commutative.
Calculating AB and BA
Reverse the order
Find BA for the same matrices.
- Use rows of B with columns of A.
- The entries are 2, 4, 1 and 5, in row order.
Answer: [2, 4; 1, 5]
Multiplying entries in the same positions is not matrix multiplication.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
Practice is filtered to your selected course. Change course or options when needed.
Check your understanding
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Which transformation acts first in AB when vectors are columns?