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Cramer's
Rule

Master the elegant determinant-based method for solving systems of linear equations with precision and confidence.

Linear Algebra

What is a System of Linear Equations?

A system of linear equations is a collection of two or more equations involving the same set of variables. The goal is to find values that satisfy all equations simultaneously.

You've likely solved these by substitution or elimination. Cramer's Rule offers a third path: using determinants.

The Determinant: Your Key Tool

Every square matrix has a number called its determinant. For a \(2 \times 2\) matrix, the determinant is computed as:

\(a\)
\(b\)
\(c\)
\(d\)
\(ad\)
\(-bc\)

Why Determinants Matter

The determinant tells us something deep about a system of equations: whether it has a unique solution, no solution, or infinitely many solutions.

\(\det \neq 0\)
The system has exactly one unique solution. Cramer's Rule applies here.
\(\det = 0\)
The system has no solution or infinitely many. Cramer's Rule does not apply.

Cramer's Rule for \(2 \times 2\) Systems

Given the system \(\;a_1 x + b_1 y = c_1\;\) and \(\;a_2 x + b_2 y = c_2\;\), define three determinants:

\(D\)
\(\det\begin{pmatrix}a_1 & b_1 \\ a_2 & b_2\end{pmatrix}\)
Coefficient matrix
\(D_x\)
\(\det\begin{pmatrix}c_1 & b_1 \\ c_2 & b_2\end{pmatrix}\)
Replace \(x\)-column with constants
\(D_y\)
\(\det\begin{pmatrix}a_1 & c_1 \\ a_2 & c_2\end{pmatrix}\)
Replace \(y\)-column with constants

Extending to \(3 \times 3\) Systems

For a system of three equations in three unknowns, the same principle applies. Given \(A\mathbf{x} = \mathbf{b}\):

For a \(3 \times 3\) determinant, you can use cofactor expansion (expansion along any row or column) or the Rule of Sarrus.

Cramer's Rule vs. Other Methods

Cramer's Rule Elimination / Substitution
Approach Compute determinants Manipulate equations directly
Best for Small systems (\(2\times2\), \(3\times3\)) Any size system
Requires \(\det(A) \neq 0\) No special condition
Find one variable? Yes: just compute that \(D_i\) Must often solve whole system
Large systems Computationally expensive More efficient

How to Apply Cramer's Rule

1
Write the system in standard form: \(a_1 x + b_1 y = c_1\), etc. All variables on the left, constants on the right.
2
Form the coefficient matrix \(A\) and compute \(D = \det(A)\). If \(D = 0\), stop: Cramer's Rule does not apply.
3
Build each modified matrix \(A_i\) by replacing the \(i\)-th column of \(A\) with the constant column \(\mathbf{b}\).
4
Compute \(D_i = \det(A_i)\) for each variable you need.
5
Divide: \(x_i = D_i / D\). Simplify your answer and verify by plugging back in.

Example 1: A \(2 \times 2\) System

1
Find \(D\): \(\det\begin{pmatrix}2 & 3 \\ 4 & -1\end{pmatrix} = (2)(-1) - (3)(4) = -2 - 12 = \) \(-14\)
2
Find \(D_x\): \(\det\begin{pmatrix}8 & 3 \\ 2 & -1\end{pmatrix} = (8)(-1) - (3)(2) = -8 - 6 = \) \(-14\)
3
Find \(D_y\): \(\det\begin{pmatrix}2 & 8 \\ 4 & 2\end{pmatrix} = (2)(2) - (8)(4) = 4 - 32 = \) \(-28\)

Visualizing the Solution

The two lines intersect at the point \((1, 2)\), confirming our Cramer's Rule result.

x y 0 1 2 3 1 2 3
\(2x+3y=8\)
\(4x-y=2\)
 (1, 2)

Example 2: Fractional Answers

D
\(\det\begin{pmatrix}3&5\\2&-1\end{pmatrix} = (3)(-1)-(5)(2) = -3-10 = \) \(-13\)
Dx
\(\det\begin{pmatrix}7&5\\4&-1\end{pmatrix} = (7)(-1)-(5)(4) = -7-20 = \) \(-27\)
Dy
\(\det\begin{pmatrix}3&7\\2&4\end{pmatrix} = (3)(4)-(7)(2) = 12-14 = \) \(-2\)

Example 3: A \(3 \times 3\) System

The coefficient matrix and its determinant:

Since \(D = 25 \neq 0\), the system has a unique solution. We proceed to find \(D_x\), \(D_y\), and \(D_z\).

Example 3: Finding Each Variable

\(D_x\)
\(\det\begin{pmatrix}9&2&1\\8&-1&3\\3&1&-1\end{pmatrix}\)

\(= 9(1-3)-2(-8-9)+1(8+3)\)
\(= -18+34+11 = 27\)
\(D_y\)
\(\det\begin{pmatrix}1&9&1\\2&8&3\\3&3&-1\end{pmatrix}\)

\(= 1(-8-9)-9(-2-9)+1(6-24)\)
\(= -17+99-18 = 64\)
\(D_z\)
\(\det\begin{pmatrix}1&2&9\\2&-1&8\\3&1&3\end{pmatrix}\)

\(= 1(-3-8)-2(6-24)+9(2+3)\)
\(= -11+36+45 = 70\)

The Column Replacement Pattern

The heart of Cramer's Rule is the column replacement pattern. Here's a visual summary for a \(2 \times 2\) system:

\(D\)
\(a_1\)
\(b_1\)
\(a_2\)
\(b_2\)
 →
\(D_x\)
\(c_1\)
\(b_1\)
\(c_2\)
\(b_2\)
 →
\(D_y\)
\(a_1\)
\(c_1\)
\(a_2\)
\(c_2\)
 Original coefficients Replace x-column with constants Replace y-column with constants

Common Mistakes

✕ Wrong Column Replaced
Replacing the \(y\)-column when solving for \(x\), or vice versa. Always match the variable to its column position.
✕ Sign Errors in Determinants
Forgetting that \(\det = ad - bc\), not \(ad + bc\). The subtraction is essential. Double-check signs.
✕ Applying When \(D = 0\)
If \(D = 0\), dividing by zero is undefined. The system is either inconsistent or dependent: Cramer's Rule cannot help.

Summary of Cramer's Rule

Concept \(2 \times 2\) System \(3 \times 3\) System
Variables \(x, y\) \(x, y, z\)
Determinants needed 3: \(D\), \(D_x\), \(D_y\) 4: \(D\), \(D_x\), \(D_y\), \(D_z\)
Formula for each variable \(x_i = D_i / D\) \(x_i = D_i / D\)
Condition \(D \neq 0\) \(D \neq 0\)
Operations per det 2 multiplications, 1 subtraction Cofactor expansion (more involved)

Question 1 of 5

For the system \(\;x + 2y = 5\;\) and \(\;3x + 4y = 6\;\), what is the value of \(D\) (the determinant of the coefficient matrix)?

Question 2 of 5

Using Cramer's Rule on \(\;x + 2y = 5\;\) and \(\;3x + 4y = 6\;\), what is \(x\)?

Question 3 of 5

If the determinant of the coefficient matrix \(D = 0\), what can we conclude?

Question 4 of 5

To find \(y\) using Cramer's Rule in a \(2 \times 2\) system, which column of the coefficient matrix do you replace with the constants?

Question 5 of 5

For the system \(\;5x - 3y = 1\;\) and \(\; -10x + 6y = 7\;\), what happens when you try to apply Cramer's Rule?