Cramer's Rule

Overview

Cramer's Rule provides an explicit formula for solving systems of linear equations using determinants. While computationally intensive for large systems, it's useful for theoretical purposes and small systems.

Prerequisites

• The system must be square ($n$ equations, $n$ unknowns)
• The coefficient matrix $A$ must be invertible ($\det(A) \neq 0$)

The Formula

For the system $A\mathbf{x} = \mathbf{b}$ where $A$ is $n \times n$:

$x_i = \frac{\det(A_i)}{\det(A)}$

where $A_i$ is $A$ with column $i$ replaced by $\mathbf{b}$.

2×2 System

For:

$a_1 x + b_1 y = c_1$ $a_2 x + b_2 y = c_2$

Solutions:

$x = \frac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}, \qquad y = \frac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}$

Example: 2×2 System

Solve:

$3x + 2y = 7$ $x - y = 1$

Step 1: Calculate $\det(A)$

$\det(A) = \begin{vmatrix} 3 & 2 \\ 1 & -1 \end{vmatrix} = 3(-1) - 2(1) = -5$

Step 2: Calculate $\det(A_1)$ - replace column 1 with $\mathbf{b}$

$\det(A_1) = \begin{vmatrix} 7 & 2 \\ 1 & -1 \end{vmatrix} = 7(-1) - 2(1) = -9$

Step 3: Calculate $\det(A_2)$ - replace column 2 with $\mathbf{b}$

$\det(A_2) = \begin{vmatrix} 3 & 7 \\ 1 & 1 \end{vmatrix} = 3(1) - 7(1) = -4$

Step 4: Apply Cramer's Rule

$x = \frac{\det(A_1)}{\det(A)} = \frac{-9}{-5} = \frac{9}{5} = 1.8$

$y = \frac{\det(A_2)}{\det(A)} = \frac{-4}{-5} = \frac{4}{5} = 0.8$

3×3 System

For three unknowns $x$, $y$, $z$:

$x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}, \quad z = \frac{\det(A_z)}{\det(A)}$

where $A_x$, $A_y$, $A_z$ have the $x$, $y$, $z$ columns replaced by $\mathbf{b}$.

Example: 3×3 System

Solve:

$x + 2y + z = 4$ $2x + y + z = 5$ $x + y + 2z = 6$

$A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & 2 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$

$\det(A) = 1(1 \times 2 - 1 \times 1) - 2(2 \times 2 - 1 \times 1) + 1(2 \times 1 - 1 \times 1)$ $= 1(1) - 2(3) + 1(1) = 1 - 6 + 1 = -4$

(After calculating all modified determinants)

$x = 1$, $y = 1$, $z = 2$

Advantages

• Gives explicit formula for each variable
• Useful for parametric solutions (variables in terms of parameters)
• Good for theoretical analysis
• Easy to compute for $2 \times 2$ and $3 \times 3$ systems

Limitations

• Only works for square systems with unique solutions
• Computationally expensive for large $n$ (requires $n+1$ determinants)
• Each determinant computation is $O(n!)$
• Not numerically stable for large systems

Comparison with Other Methods

Method	Better For
Cramer's Rule	Small systems, symbolic computation
Gaussian Elimination	General systems, numerical computation
Matrix Inverse	Multiple systems with same $A$
LU Decomposition	Large systems, repeated solving

When to Use

Use Cramer's Rule when:

• System is small ($2 \times 2$ or $3 \times 3$)
• Need explicit formula for solution
• Solving symbolically (parameters in coefficients)
• Theoretical derivations

Avoid when:

• System is large ($n > 4$)
• Need numerical efficiency
• System may be singular or nearly singular