Why does the order matter in matrix multiplication — isn't AB the same as BA?
No — matrix multiplication is generally non-commutative.
No — matrix multiplication is generally non-commutative. Each entry of a product is a row-times-column combination, and swapping the order changes which rows meet which columns, so AB and BA usually differ; sometimes only one of them is even defined (the sizes must match). This is unlike ordinary numbers, and it's why in matrix algebra you must carefully preserve the order — for example (AB)⁻¹ = B⁻¹A⁻¹, with the order reversed.
Key point
AB ≠ BA in general — matrix multiplication is non-commutative, so order is essential.
Common mistake
assuming you can freely swap the order of matrix factors.
Memory tip
keep the order; even the inverse of a product reverses it: (AB)⁻¹ = B⁻¹A⁻¹.
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