[Verse 1]
In a matrix lives a secret number
Hidden deep in rows and columns
Two by two it's simple math
A times D minus B times C
[Chorus]
Determinant, determinant
The health check of your matrix
Zero means it's broken down
Positive keeps order sound
Negative will flip around
Determinant tells the tale
[Verse 2]
When it's three by three we expand
Pick a row and dance the pattern
Plus and minus alternating
Cofactors help us calculate
Each two by two inside the frame
[Chorus]
Determinant, determinant
The health check of your matrix
Zero means it's broken down
Positive keeps order sound
Negative will flip around
Determinant tells the tale
[Bridge]
Volume scaling in disguise
Product of eigenvalues
Linear dependence exposed
When columns lose their independence
Zero screams no inverse exists
[Verse 3]
Permutations hold the key
Sign of sigma guides the way
Sum across all arrangements
Each term builds the final count
Leibniz knew this formula
[Chorus]
Determinant, determinant
The health check of your matrix
Zero means it's broken down
Positive keeps order sound
Negative will flip around
Determinant tells the tale
[Outro]
One number holds the power
Invertible or singular
Geometry meets algebra
In this magical scalar