Foundations — Cramer's rule

Every letter and squiggle in that sentence — the ingredients, the target, the "box", the "volume" — is a symbol we will now build from zero. This page assumes nothing; every notation used by the parent note is constructed here, in an order where each idea leans only on the ones before it.

---

0. What is a "system of equations", really?

Before any symbols, a picture. Two straight lines drawn on paper. Each line is one equation — every point on that line is a pair $(x,y)$ that makes that equation true. Where the two lines cross is the single pair that makes both true. That crossing point is "the solution".

Figure s01 below draws exactly this: the orange line is one equation, the teal line another, and the plum dot marks the one point $(2,1)$ that sits on both — that is what "solving the system" means.

Why "linear" matters for us: straightness is what makes the whole determinant/volume machinery work. Curved equations would break every step.

---

1. The unknowns $x_1,x_2,\dots$ and the vector $\mathbf{x}$

The little numbers we are solving for are the unknowns. We write them $x_1,x_2,\dots ,x_n$. The subscript is just a name-tag: $x_1$ is "the first unknown", $x_2$ "the second". In a 2D problem you may see them as plain $x$ and $y$.

The letter $n$ just means "however many unknowns there are". If $n=2$ we have two unknowns, if $n=3$ we have three.

---

2. Columns of a matrix, and $A$

Read a matrix column by column: each vertical strip is itself a vector — an arrow. This is the single most important way to look at $A$ for Cramer's rule.

Figure s02 takes a $2\times 2$ grid and redraws its two vertical strips as two arrows from the origin: the orange arrow is column ${\mathbf{a}}_1$, the teal arrow is column ${\mathbf{a}}_2$. The picture says "a matrix is just a bundle of arrows standing side by side."

Why the topic needs columns: Cramer's rule is entirely about swapping one column of $A$. If you don't see $A$ as a row of arrows, the whole rule looks like magic.

---

3. The product $A\mathbf{x}$ — a mix of columns

Now the crucial reinterpretation. School often teaches "multiply row by column." True — but for us the honest picture is different and far more useful:

Figure s03 shows this "tip-to-tail" recipe: the orange arrow is $x_1{\mathbf{a}}_1$, then starting from its tip we draw the teal arrow $x_2{\mathbf{a}}_2$; the plum arrow from the origin to the final tip is the whole product $A\mathbf{x}$. That plum arrow is what the mix produces.

This is why Cramer swaps a column and never a row: $A\mathbf{x}$ is literally built out of columns.

---

4. The target vector $\mathbf{b}$, and the equation $A\mathbf{x}=\mathbf{b}$

Putting it together: $\underset{\text{ingredients}}{\underbrace{A}}\underset{\text{amounts}}{\underbrace{\mathbf{x}}}=\underset{\text{target}}{\underbrace{\mathbf{b}}}.$

This compact line is the entire problem. Every worked example in the parent note is one instance of it.

---

5. What is a determinant? — signed area/volume

Here is the engine. The parent note keeps saying "$\det$ measures volume." Let's earn that picture.

Figure s04 shades the parallelogram spanned by two column-arrows ${\mathbf{a}}_1$ (orange) and ${\mathbf{a}}_2$ (teal); the shaded plum area is $|\det A|$. Widen or tilt the arrows and the shaded area — the determinant — changes with them.

Three facts about this volume are all Cramer needs — the parent note names them multilinear and alternating:

Fact 3 gives the golden test: $\det A=0$ means the columns are squashed flat (they don't span full volume), so you cannot mix them uniquely to reach a general $\mathbf{b}$. That's exactly when Cramer's rule is forbidden.

For hands-on formulas of these volumes see Determinants and Cofactor Expansion; the deeper properties live in Multilinear and Alternating Maps.

---

6. The identity matrix $I$ and basis vectors ${\mathbf{e}}_k$

The proof in the parent builds a "clever matrix" from $I$. So we need $I$.

Two properties the proof uses:

• $A{\mathbf{e}}_k={\mathbf{a}}_k$ — multiplying $A$ by the pure axis-arrow picks out column $k$ of $A$.
• $\det I=1$ — the unit box has volume $1$ (our measuring stick).

---

7. The clever matrix $X_i$ — built from $I$ and $\mathbf{x}$

The parent note's proof introduces a matrix it calls $X_i$. It is not magic — we build it here so nothing is used before it is defined.

---

8. The swapped matrix $A_i$, and the identity $A{X}_i=A_i$

Now multiply $A$ by our clever $X_i$. Matrix multiplication acts column by column: column $k$ of the product $A{X}_i$ is $A$ times column $k$ of $X_i$.

• For every wall column ($k\ne i$), that column of $X_i$ is ${\mathbf{e}}_k$, and $A{\mathbf{e}}_k={\mathbf{a}}_k$ — the original column of $A$ comes straight back.
• For the special column $i$, that column of $X_i$ is $\mathbf{x}$, and $A\mathbf{x}=\mathbf{b}$ (that is our system!).

So the product keeps every original column of $A$ except the $i$-th, which becomes $\mathbf{b}$ — that is precisely $A_i$: $A{X}_i=A_i.$

---

9. Taking determinants, and the ratio $x_i=\frac{\det A_i}{\det A}$

Now every ingredient is on the table. Take the determinant of both sides of $A{X}_i=A_i$ and use the multiplicativity rule.

Applying it to $A{X}_i=A_i$: $\det (A{X}_i)=\det A_i⟹\det A\cdot \det X_i=\det A_i.$

Why this move works: the left side is a single product $A{X}_i$, so multiplicativity splits its determinant into $\det A\cdot \det X_i$. We already earned $\det X_i=x_i$ in §7, so the $\det X_i$ factor is the unknown we want. Substituting: $\det A\cdot x_i=\det A_i.$

Finally divide by $\det A$ (allowed only when $\det A\ne 0$): $x_i=\frac{\det A_i}{\det A}.$ Top: volume of the swapped box. Bottom: volume of the original box. Divide → the pour-amount. Division demands $\det A\ne 0$ — you can never divide by a flat box.

---

Prerequisite map

See also the practical alternative Gaussian Elimination, the reformulation via Matrix Inverse, and the equivalence "$\det A\ne 0$ iff unique solution" in the Invertible Matrix Theorem.

---

Equipment checklist