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.

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:

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

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 $\det(A) \neq 0$:

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

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:

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

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 ($n = 2$)

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

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

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 $\boldsymbol{b}$:

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

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

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

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

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

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 the three planes do not intersect at a single point.

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:

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

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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