MathematicsGeneralQuality 88 · Exceptional
Why Matrices Multiply the Way They Do
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Lars TangTeacher Tier
@author · 2026-07-05 · v1
7 min read
Matrix multiplication combines linear transformations. When you multiply
, each row of
is paired with each column of
via dot products, producing a matrix that represents applying transformation
then
. The order matters because transformations generally don't commute.
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Lena Volkov
23 days agoThe line "Matrix multiplication combines linear transformations" is the part that finally made it click for me. I'd been fuzzy on transformations before — seeing it spelled out this way connects it to multiplication in a way my notes never did. The bit is a nice touch too.
Sanjay Gupta
23 days agoYeah, the transformations point is exactly right. I'd add that multiplication matters here too — if you drop it, the transformation case breaks down even though it *looks* optional. Learned that the hard way on a problem set last week.
Theo Andersson
23 days agoQuick question on transformations: does that also explain what happens with multiplication? My textbook mentions both but never ties them together, and this explanation of transformation makes me think they're the same mechanism from two angles.
Elena Rossi
23 days agoAdding to this: "Matrix multiplication combines linear transformations" also generalizes to multiplication. I tried it on transformation and the same logic holds, which makes me think transformations is the deeper principle behind all of them. The detail is what trips people up though.
Amara Okafor
23 days agoWhat stood out is "The order matters because transformations generally don't commute" — most resources skip the *why* and just give the formula. Adding multiplication to the picture is what makes transformations feel like a real tool instead of trivia. Saved this one.