Deriving the Determinant: Systems of Equations and Cramer's Rule

Goals

Show that the determinant naturally arises when solving the system of linear equations $A\boldsymbol{x} = \boldsymbol{b}$. This is historically the oldest approach, and the practical motivation is clear from the start.

Prerequisites

§1. Starting with a 2×2 System

1.1 Problem Setup

A system of two equations in two unknowns $x, y$:

\begin{cases} a_{11} x + a_{12} y = b_1 \\ a_{21} x + a_{22} y = b_2 \end{cases}

1.2 Solving for $x$: Eliminating $y$

Multiply the first equation by $a_{22}$ and the second by $a_{12}$, then subtract:

\begin{align} a_{22}(a_{11} x + a_{12} y) &= a_{22} b_1 \\ a_{12}(a_{21} x + a_{22} y) &= a_{12} b_2 \end{align}

Subtracting eliminates $y$:

$$(a_{11} a_{22} - a_{12} a_{21}) x = a_{22} b_1 - a_{12} b_2$$

1.3 The Determinant Appears

If $a_{11} a_{22} - a_{12} a_{21} \neq 0$, then:

$$x = \dfrac{a_{22} b_1 - a_{12} b_2}{a_{11} a_{22} - a_{12} a_{21}}$$

The expression in the denominator:

$$\det(A) = a_{11} a_{22} - a_{12} a_{21} = \det\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$

This is the determinant. It arises naturally from the process of solving the system!

Diagram showing how the determinant arises from elimination The Determinant Arises from Elimination a₁₁x + a₁₂y = b₁ a₂₁x + a₂₂y = b₂ System of equations Eliminate y (a₁₁a₂₂ − a₁₂a₂₁)x = a₂₂b₁ − a₁₂b₂ Coefficient of x appears det(A) = a₁₁a₂₂−a₁₂a₂₁ The Determinant! det(A) ≠ 0 ⟹ x = (a₂₂b₁ − a₁₂b₂) / det(A)
Figure 1: The determinant arises naturally from the elimination process

1.4 Solving for $y$: Eliminating $x$

Similarly, multiplying the first equation by $a_{21}$ and the second by $a_{11}$, then subtracting, eliminates $x$:

$$(a_{11} a_{22} - a_{12} a_{21}) y = a_{11} b_2 - a_{21} b_1$$

The coefficient of $y$ is the same $a_{11} a_{22} - a_{12} a_{21} = \det(A)$ as before. Dividing both sides by $\det(A) \neq 0$:

$$y = \dfrac{a_{11} b_2 - a_{21} b_1}{a_{11} a_{22} - a_{12} a_{21}}$$

1.5 The Numerators Are Also Determinants

The numerator for $x$, $a_{22} b_1 - a_{12} b_2$, can be written as a determinant:

$$a_{22} b_1 - a_{12} b_2 = \det\begin{pmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{pmatrix}$$

This is the determinant of the matrix obtained by replacing the first column of $A$ with $\boldsymbol{b}$.

Similarly, the numerator for $y$, $a_{11} b_2 - a_{21} b_1$, is also a determinant:

$$a_{11} b_2 - a_{21} b_1 = \det\begin{pmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{pmatrix}$$

This is the determinant of the matrix obtained by replacing the second column of $A$ with $\boldsymbol{b}$.

In summary:

$$x = \dfrac{\det\begin{pmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{pmatrix}}{\det(A)}, \quad y = \dfrac{\det\begin{pmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{pmatrix}}{\det(A)}$$

In both cases, we see a unified pattern: replace the column corresponding to the desired variable with $\boldsymbol{b}$. This is the essence of Cramer's rule.

§2. Cramer's Rule

2.1 The 2×2 Case

For $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ and $\boldsymbol{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix}$:

Cramer's Rule ($n = 2$)

$$x = \dfrac{\det\begin{pmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{pmatrix}}{\det(A)}, \quad y = \dfrac{\det\begin{pmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{pmatrix}}{\det(A)}$$
Cramer's rule: replacing columns of matrix A with vector b Cramer's Rule: Column Replacement Matrix A a₁₁ a₂₁ a₁₂ a₂₂ b b₁ b₂ → A₁ (for x) b₁ b₂ a₁₂ a₂₂ A₂ (for y) a₁₁ a₂₁ b₁ b₂ x = det(A₁) / det(A) y = det(A₂) / det(A) Replace col j with b → xⱼ
Figure 2: Cramer's rule — replace the j-th column of A with b to find xⱼ

2.2 Generalization to $n$ Dimensions

For an $n$-variable system $A\boldsymbol{x} = \boldsymbol{b}$, let $A_j$ denote the matrix obtained by replacing the $j$-th column of $A$ with $\boldsymbol{b}$:

Cramer's Rule (General)

$$x_j = \dfrac{\det(A_j)}{\det(A)} \quad (j = 1, 2, \ldots, n)$$

provided $\det(A) \neq 0$.

For $n = 2$, we proved this directly by elimination in Section 1. A rigorous proof for general $n$ requires cofactor expansion and the theory of adjugate matrices, which we omit here. Instead, let us verify the formula on a concrete 3×3 example.

Note: Computing 3×3 Determinants (Sarrus' Rule)

A 3×3 determinant is computed as the sum of products along downward diagonals minus the sum of products along upward diagonals:

$$\det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - afh - bdi$$
Diagram of Sarrus' rule for computing 3×3 determinants Positive terms (↘ diagonals) a b c d e f g h i +aei +bfg +cdh Negative terms (↗ diagonals) a b c d e f g h i −ceg −afh −bdi

This method works only for 3×3 matrices. For 4×4 and larger, use cofactor expansion (Chapter 4).

2.3 Worked Example: A 3×3 System

Consider the following system:

\begin{cases} x + y + z = 6 \\ 2x + 3y + z = 11 \\ x + 2y + 3z = 14 \end{cases}

The coefficient matrix and right-hand side vector:

$$A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 3 & 1 \\ 1 & 2 & 3 \end{pmatrix}, \quad \boldsymbol{b} = \begin{pmatrix} 6 \\ 11 \\ 14 \end{pmatrix}$$

First, compute $\det(A)$ by expanding along the first row:

$$\det(A) = 1 \cdot (9-2) - 1 \cdot (6-1) + 1 \cdot (4-3) = 7 - 5 + 1 = 3$$

$x$ — Replace the first column of $A$ with $\boldsymbol{b}$:

$$\det(A_1) = \det\begin{pmatrix} \color{red}{6} & 1 & 1 \\ \color{red}{11} & 3 & 1 \\ \color{red}{14} & 2 & 3 \end{pmatrix} = 6(9-2) - 1(33-14) + 1(22-42) = 3, \quad x = \dfrac{3}{3} = 1$$

$y$ — Replace the second column of $A$ with $\boldsymbol{b}$:

$$\det(A_2) = \det\begin{pmatrix} 1 & \color{red}{6} & 1 \\ 2 & \color{red}{11} & 1 \\ 1 & \color{red}{14} & 3 \end{pmatrix} = 1(33-14) - 6(6-1) + 1(28-11) = 6, \quad y = \dfrac{6}{3} = 2$$

$z$ — Replace the third column of $A$ with $\boldsymbol{b}$:

$$\det(A_3) = \det\begin{pmatrix} 1 & 1 & \color{red}{6} \\ 2 & 3 & \color{red}{11} \\ 1 & 2 & \color{red}{14} \end{pmatrix} = 1(42-22) - 1(28-11) + 6(4-3) = 9, \quad z = \dfrac{9}{3} = 3$$

Verification: Substituting $(x, y, z) = (1, 2, 3)$ into the original equations gives $1+2+3=6$, $2+6+3=11$, $1+4+9=14$, which all check out.

§3. The Case $\det(A) = 0$

3.1 Relation to Uniqueness of Solutions

When $\det(A) \neq 0$, the system $A\boldsymbol{x} = \boldsymbol{b}$ has a unique solution.

When $\det(A) = 0$:

  • No solution exists (inconsistent system), or
  • Infinitely many solutions exist (underdetermined system)

3.2 Geometric Interpretation

In 2D, $\det(A) = 0$ means the two lines are:

  • Parallel (no intersection), or
  • Coincident (infinitely many intersections)
How the value of det(A) determines the intersection of two lines Intersection of Two Lines and det(A) det(A) ≠ 0 Unique solution (intersect) det(A) = 0 No solution (parallel) det(A) = 0 Infinite solutions (coincident)
Figure 3: The relationship between det(A) and the solutions of a 2D system

In 3D, $\det(A) = 0$ means one of the following:

  • the three planes do not intersect at a single point (no solution);
  • the three planes intersect along a common line (infinitely many solutions);
  • the three planes coincide (infinitely many solutions);
  • two or more planes are parallel and do not intersect (no solution).

§4. Historical Background

4.1 Origins of the Determinant

The determinant was discovered through the process of solving systems of equations:

  • Seki Takakazu (c. 1683): Discovered in Japan while studying elimination methods for systems of equations
  • Leibniz (1693): Independently discovered in Europe
  • Cramer (1750): Formulated the general rule

4.2 Etymology

The name "determinant" comes from the fact that it determines whether a system of equations has a unique solution.

§5. Pros and Cons of This Approach

5.1 Advantages

  • Clear motivation: Starts from the natural problem "solve a system of equations"
  • Immediate usefulness: Directly connected to a solution formula
  • Historically authentic: Follows the actual path of discovery

5.2 Disadvantages

  • The formula seems to appear "out of thin air": Hard to see why it takes this particular form
  • Geometric meaning is deferred: The connection to volume requires separate explanation
  • Computationally expensive: Not practical for solving systems (Gaussian elimination is more efficient)

§6. Relation to Inverse Matrices

6.1 Inverse Matrix Formula

Expressing Cramer's rule in matrix language yields the inverse matrix formula:

$$A^{-1} = \dfrac{1}{\det(A)} \mathrm{adj}(A)$$

where $\mathrm{adj}(A)$ is the adjugate matrix (the transpose of the cofactor matrix).

6.2 $\det(A) \neq 0 \Leftrightarrow A$ Is Invertible

A matrix $A$ has an inverse (is invertible/nonsingular) if and only if $\det(A) \neq 0$.

This is another reason for the name "determinant" — it determines whether the matrix is invertible.

§7. Summary

Key Takeaways

  • The determinant was born from systems of equations: It appears naturally during the solution process
  • Cramer's rule: $x_j = \det(A_j) / \det(A)$
  • $\det(A) \neq 0$ ⇔ unique solution exists ⇔ $A$ is invertible
  • Historically the oldest approach: Seki Takakazu, Leibniz, Cramer

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