Cramer's Rule for Linear Systems

Part 18, Chapter 18: Matrices and Determinants

Learning objectives

State Cramer's rule for 2x2 and 3x3 systems
Solve a 2x2 system using Cramer's rule
Determine when Cramer's rule applies

You can stare at $A \mathbf{x} = \mathbf{b}$ for a long time and not see how to solve it. The determinant gives you an explicit closed-form answer, one short formula per unknown, expressed entirely in determinants of $A$ and modified versions of $A$ . That formula is Cramer's rule, and the closing section of this chapter shows you why it works and where it fails.

The 2x2 statement

Consider the system

$\begin{cases} ax + by = e \\ cx + dy = f \end{cases}$

Let $D = \det\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ . If $D \neq 0$ , then

$x = \frac{D_x}{D} = \frac{\det\begin{pmatrix} e & b \\ f & d \end{pmatrix}}{D}, \qquad y = \frac{D_y}{D} = \frac{\det\begin{pmatrix} a & e \\ c & f \end{pmatrix}}{D}.$

The recipe in words: to find $x$ , replace the first column of $A$ with the right-hand-side vector $(e, f)$ and take the determinant. To find $y$ , replace the second column instead. Divide each by $D$ .

Where this shows up

Circuit Analysis: Solving Kirchhoff's mesh equations for currents in a 3-loop circuit is a textbook Cramer's-rule application; the determinant in the denominator vanishes only when the circuit has redundant components.
Statics: Finding the three unknown reaction forces on a beam supported at three points is a 3x3 linear system; Cramer's rule gives a closed-form for each reaction in terms of the load distribution.
Cryptography: Hill ciphers encrypt by multiplying plaintext blocks by a key matrix; decryption requires solving the inverse system, Cramer's rule (mod 26) is one way classical cipher textbooks present this.

(Left canvas shows $A = [\mathbf{u} \mid \mathbf{v}]$ , with $\det(A) = D$ . Right canvas shows $A_x = [\mathbf{b} \mid \mathbf{v}]$ , where $\mathbf{b}$ has replaced the first column. The readout below shows $D$ , $D_x$ x, and $x = D_x / D$ x/D. Drag $\mathbf{b}$ to change the right-hand side and watch $x$ update. Toggle the radio button to compute $y$ instead.)

The 3x3 (and general) statement

For an $n \times n$ system $A \mathbf{x} = \mathbf{b}$ with $\det(A) \neq 0$ ,

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

where $A_i$ i is the matrix you get by replacing the $i$ -th column of $A$ with $\mathbf{b}$ . The pattern is identical to the $2 \times 2$ case, just more columns to track.

Why it works

The proof is a one-line application of column properties. $A \mathbf{x} = \mathbf{b}$ says the linear combination $x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \cdots + x_n \mathbf{a}_n = \mathbf{b}$ mathbfan=mathbfb. So $\mathbf{b}$ is built from the columns of $A$ with coefficients $x_i$ i. Substitute $\mathbf{b}$ for column $i$ in $A$ and use the multilinear property of $\det$ in that column: only the $x_i \mathbf{a}_i$ term survives, giving $\det(A_i) = x_i \det(A)$ det(A). Divide by $\det(A)$ .

When Cramer's rule fails

The rule requires $\det(A) \neq 0$ . If $\det(A) = 0$ , the formula has $0$ in the denominator, meaningless. Geometrically, a zero determinant means the columns of $A$ are linearly dependent, the matrix is singular, and the system either has no solution at all (inconsistent) or has infinitely many (dependent). Either way, there is no unique answer for Cramer's rule to deliver.

When to use it, when not to

Cramer's rule is conceptually elegant and theoretically powerful: it expresses the solution as an explicit ratio of determinants, which is useful for proofs and for studying how the solution depends on the right-hand side. It also works well for tiny systems $(n \leq 3)$ that you compute by hand.

For larger systems, however, it is computationally terrible. An $n \times n$ determinant via cofactor expansion costs roughly $n!$ operations; Cramer's rule needs $n+1$ of them. So a $10 \times 10$ system via Cramer's rule needs about $11 \cdot 10! \approx 40\,000\,000$ operations, while Gaussian elimination handles it in around $1000$ . Use Cramer's rule for understanding; use elimination for computation.

Try it

Predict first: solve $3x + y = 5, x + 2y = 4$ by hand, what are $x$ and $y$ ? Set $\mathbf{u} = (3, 1), \mathbf{v} = (1, 2), \mathbf{b} = (5, 4)$ in the widget and verify, then substitute back into the original system to double-check.
Solve on paper: $2x + 3y = 8$ , $x - y = 1$ . Find $D$ , $D_x$ x, $D_y$ y, then $x$ and $y$ .
Try to apply Cramer's rule to $x + 2y = 3$ , $2x + 4y = 6$ . Compute $D$ . What goes wrong? What does it tell you about the system?
For the $3 \times 3$ system $x + y + z = 6, \ 2y + z = 4, \ 3z = 9$ (already triangular), use Cramer's rule to find $x$ . Confirm against back-substitution.

Pause: Cramer's rule requires $D \neq 0$ . But $D = 0$ can mean two different things, no solution or infinitely many. How would you tell which?

A trap to watch for

The most common error: replacing the wrong column. To find $x_3$ , you replace column 3. Students sometimes replace column 1, the column that contains $x_3$ -related entries in their head, but Cramer's rule is about the position of the unknown, not the position of the coefficient. **Always: $x_i$ i means replace column $i$ .**

A second trap: applying Cramer's rule when $D = 0$ . The formula gives $\tfrac{0}{0}$ or $\tfrac{\text{nonzero}}{0}$ , both meaningless. Beginners sometimes write " $x = \tfrac{0}{0} = 0$ " and march on. The correct response when $D = 0$ is to stop and use elimination or substitution to determine whether the system has no solution or infinitely many. Cramer's rule is silent on those cases.

What you now know

You can apply Cramer's rule to any $2 \times 2$ or $3 \times 3$ linear system with a non-zero coefficient determinant, recognise when the rule fails and what that failure means geometrically, and choose between Cramer's rule and elimination depending on the size of the system. This is the closing section of the textbook. You now have the working machinery, numbers, operations, linearity, geometry, induction, summation, matrices, determinants, that supports every elementary mathematics course you will take next.

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References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 17, §6, Cramer's rule with both $2 \times 2$ and $3 \times 3$ derivations.

Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press. Chapter 5: Cramer's rule alongside the cost comparison with elimination.

Hoffman, K.; Kunze, R. (1971). Linear Algebra (2nd ed.). Prentice-Hall. §5.6, the column-replacement proof via multilinear algebra.

Axler, S. (2015). Linear Algebra Done Right (3rd ed.). Springer. Chapter 10: the determinant of the inverse and Cramer's rule in operator language.

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