[Verse 1] In the matrix world where numbers dance There's a special function, give it a chance Takes a square array and spits one number out Determinant's the name, let me tell you what it's about Multilinear in every row and column line Alternating when you swap, the sign will decline Normalized so identity gives you one Three properties make the determinant done [Chorus] Det of A times B equals det A times det B Transpose leaves it unchanged, that's the key Scale by c and raise to n, volume's what it tells Zero means it's singular, invertible it repels Determinant, determinant, the matrix oracle Determinant, determinant, geometrical [Verse 2] Two by two is easy, ad minus bc But when dimensions grow, permutations run free Sigma runs through all arrangements of the set Sign of permutation times products you get First row first sigma, second row second place All the way down with mathematical grace Sum them up and that's your answer clear The determinant formula engineers hold dear [Chorus] Det of A times B equals det A times det B Transpose leaves it unchanged, that's the key Scale by c and raise to n, volume's what it tells Zero means it's singular, invertible it repels Determinant, determinant, the matrix oracle Determinant, determinant, geometrical [Bridge] Volume scaling factor, that's the geometric view Positive preserves the orientation too Negative will flip it like a mirror's reflection Zero collapses space in at least one direction Eigenvalues multiply to give the same result Kernel's not just zero when singularity's the result [Chorus] Det of A times B equals det A times det B Transpose leaves it unchanged, that's the key Scale by c and raise to n, volume's what it tells Zero means it's singular, invertible it repels Determinant, determinant, the matrix oracle Determinant, determinant, geometrical [Outro] Linear dependence hiding in the zero case Determinant reveals the matrix true face From algebra to geometry it builds the bridge Determinant's the gatekeeper on knowledge ridge
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