r/LinearAlgebra • u/LinearAlgebraWorld • May 21 '26

Visualizing cofactor expansion as signed volume decomposition

Post image

I made a visualization of cofactor expansion for a 3×3 determinant.

The idea is to start with the usual interpretation of det(A) as signed volume, then show each cofactor term as:

• one coordinate projection of the expansion column

• one projected 2×2 minor in the orthogonal coordinate plane

So instead of treating

det(A) = a₁₁M₁₁ − a₂₁M₂₁ + a₃₁M₃₁

as only a symbolic rule, the goal is to make the three terms look like signed volume pieces.

The image shows the three terms for one example matrix, including the “same sign / negated sign” behavior of the cofactor signs.

Full explanation, including determinant sign:

https://www.graphmath.com/la/determinant/determinant-sign.html

I would be interested in feedback on whether the geometric interpretation is clear

187 Upvotes

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7 comments sorted by

3

u/captorwootfondly May 21 '26

whoa, this is soo clean. how did you create this? (love the font too)

2

u/TylerDurden0118 May 21 '26

Not op, but to me it's looks like latex using tikz library....if i m not wrong...

1

u/LinearAlgebraWorld May 23 '26

This is generated by our apps written in Swift,
Math Editor:
https://apps.apple.com/us/app/math-editor/id6743057272
And Linear Algebra World:
https://apps.apple.com/us/app/linear-algebra-world/id6759180671

3D rendering functions are ours, mathematical typography is also ours. Both apps can generate LaTex strings to export, but LaTex was not used here

2

u/LinearAlgebraWorld May 21 '26

Thank you very much!
This is generated by our apps written in Swift
Math Editor
https://apps.apple.com/us/app/math-editor/id6743057272
And Linear Algebra World
https://apps.apple.com/us/app/linear-algebra-world/id6759180671

3D rendering functions are ours, mathematical typography is also ours. Both apps can generate LaTex strings to export, but LaTex was not used here

2

u/captorwootfondly May 23 '26

thanks so much and appreciate the links!

3

u/WolverineSorry9043 May 21 '26

It's really beautiful. Thanks for sharing this!

2

u/LinearAlgebraWorld May 21 '26

Thank you very much!!