Foundations — Determinants — cofactor expansion along any row - column

Before you can read the parent note on cofactor expansion, you must own every symbol it throws at you. Below, each idea is built on the previous one — read top to bottom, skip nothing.

1. A matrix, and its size $n$

Picture it as a spreadsheet: number the rows top-to-bottom $1,2,\dots ,n$ and the columns left-to-right $1,2,\dots ,n$. Figure s01 labels these row and column indices on our example matrix.

Why the topic needs it: the determinant is only defined for square matrices, and the general formulas run over $1,2,\dots ,n$ — so you must know what $n$ counts before any sum makes sense.

2. The entry $a_{ij}$ — addressing one number

Read it like a street address: $a_{23}$ means "go to row 2, then column 3". In the matrix above, $a_{23}=5$ and $a_{31}=1$.

Why the topic needs it: every formula in the parent note multiplies specific entries $a_{ij}$; you must be able to pick the exact number out of the grid instantly.

3. The symbol $\det$ — the "determinant of" operation

We have not yet said how to compute $\det A$ for a big matrix — that is the whole point of the parent note. Here we are only fixing what the symbol means so later formulas like $\det A=\dots$ read as "the number $\det A$ equals …". The upcoming sections (§4, §5) give the actual recipes for the small cases.

Why the topic needs it: the parent note opens straight into "$\det A=\sum \dots$" — you cannot read that line unless $\det$ already means something to you.

4. Sigma notation $\sum$ — "add up a list"

Why the topic needs it: the expansion formula $\det A={\sum }_{j=1}^n{a}_{ij}{C}_{ij}$ is nothing but "walk along a line and add up the pieces". Without $\sum$ you can't read it.

5. The $1\times 1$ and $2\times 2$ determinants — the base cases

Everything the parent note does recurses down until the leftover matrices are tiny. So we must nail the smallest cases where $\det$ is defined outright, no recipe needed.

Why the topic needs it: cofactor expansion turns a $3\times 3$ into $2\times 2$ pieces, and a $4\times 4$ into $3\times 3$ into $2\times 2$ into $1\times 1$. These two tiny cases are the floor of the recursion — the recipe stops here because these are defined directly.

6. Deleting a row and column → the Minor $M_{ij}$

Why the topic needs it: the minor is the smaller determinant each step of the expansion produces. It is how the recursion physically happens — a $3\times 3$ minor-hunt gives $2\times 2$ leftovers, which give $1\times 1$ leftovers.

7. The sign $(-1{)}^{i+j}$ — the checkerboard

Why does raising $-1$ to a power give this? Because $-1$ multiplied by itself an even number of times returns to $+1$, and an odd number of times lands on $-1$. So the parity (even-or-odd-ness) of $i+j$ is all that matters.

Why the topic needs it: without this sign the smaller determinants would add up to the wrong number. The sign encodes a deeper fact (permutation parity, from the Leibniz formula for determinants) — but for computing, it is simply the checkerboard.

8. The Cofactor $C_{ij}$ — signed minor

Why the topic needs it: the entire expansion is built from entries times cofactors. The cofactor is the atom of the formula.

9. Putting the symbols together — row AND column expansion

First fix one particular row, call its number $i$ (any single value from $1$ to $n$ that you choose). Then let $j$ sweep across that row. Now every piece of the parent formula is defined:

$\det A=\underset{\text{§4 add up}}{\underbrace{\sum _{j=1}^n}}\underset{\text{§2 entry}}{\underbrace{a_{ij}}}\underset{\text{§7 sign}}{\underbrace{(-1{)}^{i+j}}}\underset{\text{§6 minor}}{\underbrace{M_{ij}}}(\text{row }i\text{ fixed})$

Read it aloud: "Pick one row $i$. Walk along that row; for each entry, multiply by its checkerboard sign and its minor; add them all."

10. How these foundations feed the topic

Each box on the left must be solid before the "Cofactor expansion" box on the right makes sense.

Equipment checklist

Give the answer in your head, then reveal:

Connections

• Leibniz formula for determinants — the deep source of the checkerboard sign (permutation parity).
• Properties of determinants (multilinearity, alternation) — why deleting/adding rows behaves the way it does.
• Determinant via row reduction — the faster cousin once you know these basics.
• Adjugate matrix and inverse — a whole matrix built out of cofactors.
• Cramer's rule — solving equations with cofactors.
• Invertibility and singular matrices — $\det A\ne 0⟺$ invertible.