Determinant: Expansion via Linearity and Alternation | Generated by AI
When you learn that the determinant is a function that is
- multilinear in the rows (or columns), and
- alternating (swapping two rows flips its sign),
the cofactor‐expansion formula falls out naturally. Here’s a step‐by‐step “why” for the 3×3 case:
1. Determinant via permutations
The most general definition of a 3×3 determinant is
\[\det A \;=\; \sum_{\sigma\in S_3}\!\mathrm{sgn}(\sigma)\,a_{1,\sigma(1)}\,a_{2,\sigma(2)}\,a_{3,\sigma(3)},\]where you sum over all six permutations σ of {1,2,3}, and sgn(σ) is +1 for even permutations and –1 for odd ones.
If you write them out, you get exactly the six terms:
\[a_{11}a_{22}a_{33},\;a_{12}a_{23}a_{31},\;a_{13}a_{21}a_{32} \;-\; \bigl(a_{13}a_{22}a_{31}+a_{11}a_{23}a_{32}+a_{12}a_{21}a_{33}\bigr).\]2. Grouping by one row or column
Notice you can group those six terms according to which entry in the first row they use:
-
Terms containing $a_{11}$:
\[a_{11}\,(a_{22}a_{33} - a_{23}a_{32}) = a_{11}\det\!\begin{pmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{pmatrix}.\] -
Terms containing $a_{12}$:
\[a_{12}\,(a_{23}a_{31} - a_{21}a_{33}) = -\,a_{12}\det\!\begin{pmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{pmatrix}.\] -
Terms containing $a_{13}$:
\[a_{13}\,(a_{21}a_{32} - a_{22}a_{31}) = \;a_{13}\det\!\begin{pmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}.\]
That’s exactly the cofactor expansion along the first row:
\[\det A = a_{11}C_{11} \;+\; a_{12}C_{12} \;+\; a_{13}C_{13},\]where each cofactor $C_{1j} = (-1)^{1+j}\det(\text{minor}_{1j})$.
3. Why “minors” and “signs”?
- Minor: $\det(\text{minor}_{ij})$ is the determinant of the 2×2 submatrix you get by deleting row i and column j.
- Sign factor $(−1)^{i+j}$: ensures that if you swap two rows of $A$, the overall sign flips—just like the permutation definition demands.
Because the determinant must be linear in each row, you can “pull out” one entry from a row, leaving the determinant of the smaller matrix multiplied by that entry. The alternating property forces those $(−1)$ signs.
4. Geometric intuition
For a 3×3 matrix whose rows are vectors $\mathbf{u},\mathbf{v},\mathbf{w}\in\mathbb{R}^3$, $\det A$ is the signed volume of the parallelepiped they span.
- Expanding along, say, the first row, you’re decomposing that volume into “slabs” perpendicular to each coordinate axis.
- Each slab’s area is the area of the parallelogram spanned by the other two row‐vectors (a 2×2 determinant), and you scale it by the corresponding coordinate $u_j$ of $\mathbf{u}$.
- The sign $(−1)^{1+j}$ keeps track of orientation.
5. Generalization to $n\times n$
All of this works in the same way for any size $n$: the permutation definition has $n!$ terms, you group them by one row or column, and each group collapses to
\[a_{ij}\,(-1)^{i+j}\det(\text{minor}_{ij}).\]That is the cofactor expansion principle.
In short: Cofactor expansion is simply the permutation‐sum definition of the determinant, reorganized by one row (or column), together with the requirements of linearity and sign‐alternation. It offers a recursive way to compute—and a clear geometric picture of—the determinant.