Difficulty: Intermediate | Prerequisites: 3×3 determinants, matrix notation for linear systems
Cramer's rule gives an explicit formula for each variable in a linear system Ax = b, expressed entirely in terms of determinants. It is most useful for small systems (2×2 or 3×3) and for theoretical results. For larger systems, row reduction is more efficient in practice, but Cramer's rule is a staple of exam questions and provides a neat connection between determinants and solutions.
Cramer's rule solves the system Ax = b by computing one determinant per variable. Each variable xᵢ equals the determinant of the matrix formed by replacing column i of A with the vector b, divided by det(A). It works only when det(A) ≠ 0.
Cramer's rule
A formula that gives the unique solution to a system Ax = b when A is an invertible n×n matrix. Each variable xᵢ = det(Aᵢ) / det(A). Think of it as solving one variable at a time using determinants instead of row-reducing the whole augmented matrix.
Coefficient matrix (A)
The n×n matrix formed by the coefficients of the variables in the system of equations. It does not include the constants on the right-hand side.
Replacement matrix (Aᵢ)
The matrix obtained by replacing the i-th column of A with the right-hand side vector b. For a 3×3 system, A₁ replaces column 1 with b, A₂ replaces column 2 with b, and A₃ replaces column 3 with b.
Right-hand side vector (b)
The column vector of constants on the right side of the system Ax = b.
Given a system Ax = b where A is n×n and det(A) ≠ 0:
xᵢ = det(Aᵢ) / det(A)
where Aᵢ is A with column i replaced by b.
For a 3×3 system, you compute four determinants in total: det(A), det(A₁), det(A₂), and det(A₃).
Write the system in the form Ax = b. Identify A (the coefficient matrix) and b (the right-hand side).
Compute det(A). If det(A) = 0, Cramer's rule does not apply (the system either has no solution or infinitely many).
For each variable xᵢ, form Aᵢ by replacing column i of A with b.
Compute det(Aᵢ).
Divide: xᵢ = det(Aᵢ) / det(A).
System: A = [−1, 1, 0; −3, 0, 1; 0, 1, 2], b = [7, −8, −3].
det(A) = 8 (computed by cofactor expansion along row 1, verifying it is non-zero).
Finding x₁:
A₁ = [7, 1, 0; −8, 0, 1; −3, 1, 2]. det(A₁) = 6.
x₁ = 6/8 = 3/4.
Finding x₂:
A₂ = [−1, 7, 0; −3, −8, 1; 0, −3, 2]. det(A₂) = 68.
x₂ = 68/8 = 17/2.
Finding x₃:
A₃ = [−1, 1, 7; −3, 0, −8; 0, 1, −3]. det(A₃) = −46.
x₃ = −46/8 = −23/4.
Solution: x₁ = 3/4, x₂ = 17/2, x₃ = −23/4.
System: A = [−1, 1, 1; −1, 0, 2; 3, 1, 3], b = [3, 7, −3].
det(A) = 4.
det(A₁) = −26, so x₁ = −26/4 = −13/2.
det(A₂) = 63, so x₂ = 63/4.
det(A₃) = 1, so x₃ = 1/4.
Solution: x₁ = −13/2, x₂ = 63/4, x₃ = 1/4.
Cramer's rule is used in theoretical economics to show how equilibrium prices change when supply or demand shifts (comparative statics). In engineering, it provides closed-form solutions for small circuit or structural analysis problems where you need an algebraic expression for a single unknown rather than a numerical answer.
Students sometimes replace a row of A with b instead of a column. Cramer's rule always replaces a column.
Forgetting to check det(A) ≠ 0 before applying Cramer's rule. If det(A) = 0, the rule does not work and the system may have no solution or infinitely many solutions.
Arithmetic errors in the replacement determinants are the most common source of wrong answers. Each det(Aᵢ) is a full 3×3 determinant, so the computation is substantial. Double-check each one.
Students occasionally divide det(A) by det(Aᵢ) instead of the other way around. The formula is xᵢ = det(Aᵢ) / det(A), not the reverse.
⚠️ A full Cramer's rule problem for a 3×3 system requires four 3×3 determinants. Budget your time: each one is a cofactor expansion.
⚠️ "Use Cramer's rule to find the solution" is a directive. Even if row reduction would be faster, you must use Cramer's rule when the problem says so.
⚠️ If you are asked only for x₂, you need only det(A) and det(A₂), not all four determinants.
True or false: Cramer's rule works when det(A) = 0.
False.
Fill in the blank: To find x₂ using Cramer's rule, replace column ___ of A with b.
Column 2.
True or false: In Cramer's rule, xᵢ = det(A) / det(Aᵢ).
False. It is det(Aᵢ) / det(A).
Fill in the blank: For a 3×3 system, Cramer's rule requires computing ___ determinants in total.
Four (det(A), det(A₁), det(A₂), det(A₃)).
Q: Using Cramer's rule, solve the system with A = [−1, 1, 0; −3, 0, 1; 0, 1, 2] and b = [7, −8, −3].
A: det(A) = 8. det(A₁) = 6, det(A₂) = 68, det(A₃) = −46. Solution: x₁ = 3/4, x₂ = 17/2, x₃ = −23/4.
Q: You are given a 3×3 system where det(A) = 4 and det(A₂) = 63. What is x₂?
A: x₂ = det(A₂)/det(A) = 63/4.
Q: Why must you check det(A) before applying Cramer's rule?
A: If det(A) = 0, the coefficient matrix is singular and the system either has no unique solution or infinitely many solutions. Division by zero is undefined, so the formula breaks down.
Q: In Cramer's rule, which part of A do you replace to find x₃?
A: Replace column 3 of A with the right-hand side vector b to form A₃.
Q: A 3×3 system has det(A) = 4, det(A₁) = −26, det(A₂) = 63, det(A₃) = 1. State the full solution.
A: x₁ = −26/4 = −13/2, x₂ = 63/4, x₃ = 1/4.
This connects to determinants because the entire method is built on determinant computation. Speed and accuracy with 3×3 determinants is a prerequisite.
This connects to matrix inverses because Cramer's rule is equivalent to x = A⁻¹b, just expressed in determinant form rather than as a matrix product.
This connects to the broader theory of linear systems: Cramer's rule applies only when the system has a unique solution (det(A) ≠ 0), which is the same condition for A being invertible.
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