Foundations — Determinant of 3×3 matrix — cofactor expansion

This page assumes nothing. Before you can read the parent note 2.6.8, you need to be fluent in a small pile of symbols and pictures. We build every single one from the ground up, in the order they depend on each other.

1. What a matrix is, and what the little numbers mean

$A=\left[\begin{array}{ccc}2& -3& 1\\ 5& 0& 6\\ -1& 4& 2\end{array}\right]$

This one has 3 rows and 3 columns, so we call it a 3×3 matrix (say "three by three": rows first, then columns).

The picture: think of a spreadsheet, or seats in a small cinema — 3 rows deep, 3 seats wide.

Now, how do we point at one specific number? We use a subscript — the small numbers written low and to the right of a letter.

Look at the figure. The entry $a_{23}$ is in row 2, column 3 — that's the $6$ in matrix $A$ above. The entry $a_{31}$ is row 3, column 1 — that's the $-1$.

Why the topic needs this: cofactor expansion talks constantly about "delete row $i$ and column $j$" and "the sign $(-1{)}^{i+j}$". If you can't instantly locate $a_{ij}$, none of it makes sense.

2. The two vertical bars — determinant notation

You will see the same grid of numbers written two different ways:

$\left[\begin{array}{cc}2& -3\\ 5& 0\end{array}\right]\text{versus}\left|\begin{array}{cc}2& -3\\ 5& 0\end{array}\right|$

We also write the determinant as $\det (A)$. So these three all mean the same single number:

$\det (A)=\left|A\right|=\left|\begin{array}{ccc}2& -3& 1\\ 5& 0& 6\\ -1& 4& 2\end{array}\right|$

The picture: brackets = the box of ingredients; bars = the one number you bake out of them.

Why the topic needs this: every minor $M_{ij}$ is written with vertical bars — it is itself a determinant, not a matrix. Confusing the two makes the whole formula look like nonsense. See 2.6.01-What-is-a-determinant for the deeper "what does this number mean" story.

3. The 2×2 determinant — the atom of everything

Everything in the parent note eventually reduces to computing 2×2 determinants, so we must nail this cold.

What this says in words: multiply the two numbers on the main diagonal (top-left to bottom-right: $a$ and $d$), then subtract the product of the two on the anti-diagonal (top-right to bottom-left: $b$ and $c$).

Why "$ad-bc$" and not "$ad+bc$"? The determinant of a 2×2 is the signed area of the parallelogram made by the two rows (thought of as arrows from the origin). The subtraction is what makes the area come out negative when the two arrows are swapped — it records orientation (which arrow is "on the left"). Look at the amber parallelogram in the figure: its area is exactly $|ad-bc|$, and the sign tells you if the second arrow is clockwise or counter-clockwise from the first. Full story in 2.6.05-Determinant-of-2×2-matrix.

Why the topic needs this: cofactor expansion turns one 3×3 determinant into three of these 2×2 determinants. This is the tool the whole method is built on.

4. Deleting a row and column — the minor $M_{ij}$

Here is the central move of cofactor expansion, and it is purely visual.

Follow the figure. To find $M_{12}$ (minor of position row 1, column 2):

1. Put a finger on row 1 — cross the whole row out.
2. Put a finger on column 2 — cross the whole column out.
3. Four numbers survive, in the corners. Read them off in their original positions as a 2×2 matrix.
4. Take its determinant.

For our matrix $A$: $M_{12}=\left|\begin{array}{cc}5& 6\\ -1& 2\end{array}\right|=(5)(2)-(6)(-1)=10+6=16$

Why the topic needs this: minors are the "smaller pieces we already know how to compute." Every term in the expansion is (a number) × (a sign) × (a minor).

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

The last symbol is a small power of $-1$.

Why? Multiplying $-1$ by itself an even number of times cancels back to $+1$; an odd number of times leaves one $-1$. Example: $(-1{)}^{1+2}=(-1{)}^3=-1$; and $(-1{)}^{2+2}=(-1{)}^4=+1$.

You never need to compute the power in practice. Just memorise the checkerboard, where the top-left is always $+$:

$\left[\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right]$

The picture: a chessboard. The corner square is white (+), and the colour flips every time you step one square sideways or up/down.

So a cofactor is just a minor wearing its correct plus or minus.

Why the topic needs this: the signs are not decoration — they record orientation (positive vs negative volume), exactly like the minus in $ad-bc$ did for area. Drop them and you get the wrong number every time.

6. The Greek letters and the sum symbol

The parent note's derivation uses three pieces of "scary-looking" shorthand. Here is all of it, demystified.

It's just a compact way to write "do this for each column and total it."

The parent note's proof section also mentions $\sigma$ (a permutation — a reshuffling of the columns $1,2,3$) and $\text{sgn}(\sigma )$ (its sign, $+1$ or $-1$). You do not need these to use cofactor expansion — they only appear in the "why does this work" derivation. If they intimidate you, skip that block on first read; the recipe stands on its own.

7. Linear dependence — why some determinants are zero

The parent note's Example 3 uses the phrase "linearly dependent rows" to explain a zero determinant. Here is the plain-words picture.

Why the topic needs this: it explains when and why $\det =0$ happens — the single most important special case. More in 2.6.09-Properties-of-determinants.

Prerequisite map

Every arrow means "you need the left box before the right box makes sense." The topic sits at the bottom; each foundation on this page feeds into it.

Where these go next

• The number itself and its meaning: 2.6.01-What-is-a-determinant
• The 2×2 atom in full: 2.6.05-Determinant-of-2×2-matrix
• Shortcuts that exploit zeros and dependence: 2.6.09-Properties-of-determinants
• Using determinants to invert matrices: 2.6.11-Determinant-and-matrix-inverse
• Solving equations with them: 2.7.03-Cramers-rule
• The 3D geometry link: 3.4.02-Cross-product-and-determinants

Equipment checklist

Test yourself — say the answer out loud, then reveal.