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Cramer's rule

Cramer's rule is a theorem in linear algebra that provides an explicit formula for solving a system of n linear equations in n unknowns using determinants of matrices formed from the system's coefficients and constants, applicable only when a unique solution exists.^[1] Formulated by the Swiss mathematician Gabriel Cramer (1704–1752), the rule was first published in 1750 in his book Introduction à l'analyse des lignes courbes algébriques, where it appears in Chapter 3 as a general method for solving linear systems using determinants, applicable to any n.^[2] For a system Ax = b, where A is the n × n coefficient matrix, x is the vector of unknowns, and b is the constant vector, Cramer's rule states that the i-th component of the solution is given by x_i = det(A_i) / det(A), with A_i obtained by replacing the i-th column of A with b, provided det(A) ≠ 0 to ensure invertibility and uniqueness.^[1] This determinant-based approach highlights the connection between linear systems and matrix theory, predating modern vector spaces but integral to their development. While elegant for theoretical proofs and small systems—such as in circuit analysis or economic modeling—the rule's computational complexity, requiring the evaluation of n+1 determinants each of O(n^3) complexity with standard algorithms, for an overall O(n^4) cost, renders it inefficient for large n compared to methods like Gaussian elimination. Despite this, it remains a pedagogical cornerstone for illustrating the role of determinants in linear algebra.^[3]

Overview and Formulation

Historical Context

Cramer's rule originated from the work of the Swiss mathematician Gabriel Cramer (1704–1752), who developed the method as part of his contributions to algebraic geometry.^[2] Earlier related ideas on solving systems of linear equations through combinatorial expansions akin to determinants appeared in the 17th century, notably in the correspondence of Gottfried Wilhelm Leibniz (1646–1716), who described elimination techniques and "resultants" for such systems in 1693.^[4] Cramer presented the rule in full generality for an arbitrary number of unknowns in his 1750 publication Introduction à l'analyse des lignes courbes algébriques, where it was presented in Chapter 3, with details for systems of four equations in the appendix, as a tool for solving linear systems arising in the analysis of algebraic curves.^[2]^[5] The rule's recognition grew in the 19th century alongside the formal development of determinant theory, which Augustin-Louis Cauchy (1789–1857) advanced by introducing the term "determinant" and its properties in 1812, facilitating its integration into linear algebra.^[4] By mid-century, it became a standard topic in European mathematics textbooks on algebra and geometry, reflecting the era's emphasis on systematic equation solving.^[4]

General Statement

Cramer's rule provides an explicit formula for the unique solution to a system of n linear equations in n unknowns when the coefficient matrix is invertible.^[6] Consider the system Ax = b, where A is an n \times n matrix, x = (x_1, \dots, x_n)^T is the column vector of unknowns, and b is a given column vector in \mathbb{R}^n.^[6] The matrix A may be denoted as A = [\mathbf{a}_1, \dots, \mathbf{a}_n], with \mathbf{a}_i denoting the i-th column vector of A.^[6] The prerequisite for the rule's applicability is that A must be invertible, which holds if and only if \det(A) \neq 0; under this condition, the system has a unique solution.^[6] The component-wise formula for the solution is then given by

x_i = \frac{\det(A_i)}{\det(A)}, \quad i = 1, \dots, n,

where A_i is the matrix obtained by replacing the i-th column of A with the vector b.^[6] In vector form, the solution satisfies x = A^{-1} b, and Cramer's rule expresses this equivalently through the determinant ratios for each entry of x.^[6] The rule is named after the Swiss mathematician Gabriel Cramer.^[7]

Practical Usage

Computation Steps

To apply Cramer's rule for solving a system of linear equations Ax = b, where A is an n \times n coefficient matrix and b is the constant vector, the procedure involves systematic determinant evaluations.^[8] The first step is to compute the determinant of the coefficient matrix A, denoted \det(A). If \det(A) = 0, the matrix is singular, and the system lacks a unique solution, so Cramer's rule cannot be applied. The second step requires, for each index i from 1 to n, constructing a modified matrix A_i by replacing the i-th column of A with the constant vector b, and then computing \det(A_i). This process generates n such modified matrices and their determinants.^[8] In the third step, each component of the solution vector is obtained as x_i = \det(A_i) / \det(A) for i = 1 to n. These ratios directly yield the explicit solution components. Determinant calculations in these steps are typically performed using cofactor expansion, which recursively applies the Laplace expansion formula along a row or column; this method is efficient and explicit for small dimensions (n \leq 3), where closed-form expressions or simple expansions suffice, but it is not recommended for detailed algorithmic implementation beyond that. Overall, the procedure demands n+1 determinant computations, each with a time complexity of O(n!) under cofactor expansion, rendering Cramer's rule computationally prohibitive for systems where n > 3 due to the factorial growth in operations.^[9]

Illustrative Examples

To illustrate Cramer's rule, consider a simple 2×2 system of linear equations:

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

The coefficient matrix is A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, with determinant \det(A) = ad - bc. Assuming \det(A) \neq 0, the solution is given by x = \frac{\det(A_x)}{\det(A)} and y = \frac{\det(A_y)}{\det(A)}, where A_x = \begin{pmatrix} e & b \\ f & d \end{pmatrix} so \det(A_x) = ed - bf, and A_y = \begin{pmatrix} a & e \\ c & f \end{pmatrix} so \det(A_y) = af - ec. Thus, x = \frac{ed - bf}{ad - bc} and y = \frac{af - ec}{ad - bc}.^[8] For a concrete 2×2 example, take a=3, b=2, e=8, c=1, d=4, f=7. Then \det(A) = 3 \cdot 4 - 2 \cdot 1 = 10, \det(A_x) = 8 \cdot 4 - 2 \cdot 7 = 18, and \det(A_y) = 3 \cdot 7 - 8 \cdot 1 = 13, yielding x = 18/10 = 1.8 and y = 13/10 = 1.3.^[8] Now consider a 3×3 system to demonstrate the method for larger matrices:

\begin{cases} 2x + y + 0z = 5 \\ x + 3y + z = 6 \\ 0x + y + 2z = 3 \end{cases}

The coefficient matrix is A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 2 \end{pmatrix}, and the constant vector is \mathbf{b} = \begin{pmatrix} 5 \\ 6 \\ 3 \end{pmatrix}. First, compute \det(A) using cofactor expansion along the third column (which has a zero for simplicity):

\det(A) = 0 \cdot C_{13} + 1 \cdot C_{23} + 2 \cdot C_{33},

where C_{ij} = (-1)^{i+j} \det(M_{ij}) and M_{ij} is the minor matrix. Here, C_{23} = (-1)^{2+3} \det\begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} = - (2 \cdot 1 - 1 \cdot 0) = -2, and C_{33} = (-1)^{3+3} \det\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} = (6 - 1) = 5. Thus, \det(A) = 1 \cdot (-2) + 2 \cdot 5 = 8. Next, form A_x by replacing the first column of A with \mathbf{b}:

A_x = \begin{pmatrix} 5 & 1 & 0 \\ 6 & 3 & 1 \\ 3 & 1 & 2 \end{pmatrix}.

Expand \det(A_x) along the third column:

\det(A_x) = 0 \cdot C_{13}' + 1 \cdot C_{23}' + 2 \cdot C_{33}',

with C_{23}' = (-1)^{2+3} \det\begin{pmatrix} 5 & 1 \\ 3 & 1 \end{pmatrix} = - (5 \cdot 1 - 1 \cdot 3) = -2, and C_{33}' = (-1)^{3+3} \det\begin{pmatrix} 5 & 1 \\ 6 & 3 \end{pmatrix} = (15 - 6) = 9. Thus, \det(A_x) = 1 \cdot (-2) + 2 \cdot 9 = 16, so x = 16/8 = 2. For y, replace the second column:

A_y = \begin{pmatrix} 2 & 5 & 0 \\ 1 & 6 & 1 \\ 0 & 3 & 2 \end{pmatrix}.

Expand along the third column:

\det(A_y) = 0 \cdot C_{13}'' + 1 \cdot C_{23}'' + 2 \cdot C_{33}'',

with C_{23}'' = (-1)^{2+3} \det\begin{pmatrix} 2 & 5 \\ 0 & 3 \end{pmatrix} = - (6 - 0) = -6, and C_{33}'' = (-1)^{3+3} \det\begin{pmatrix} 2 & 5 \\ 1 & 6 \end{pmatrix} = (12 - 5) = 7. Thus, \det(A_y) = 1 \cdot (-6) + 2 \cdot 7 = 8, so y = 8/8 = 1. Finally, for z, replace the third column:

A_z = \begin{pmatrix} 2 & 1 & 5 \\ 1 & 3 & 6 \\ 0 & 1 & 3 \end{pmatrix}.

Expand along the third column:

\det(A_z) = 5 \cdot C_{13}''' + 6 \cdot C_{23}''' + 3 \cdot C_{33}''',

with C_{13}''' = (-1)^{1+3} \det\begin{pmatrix} 3 & 6 \\ 1 & 3 \end{pmatrix} ? No, wait: M_{13} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}, \det=1, C_{13}=(+1) \cdot 1 = 1; M_{23} = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}, \det=2, C_{23}=(-1) \cdot 2 = -2; M_{33} = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}, \det=5, C_{33}=(+1) \cdot 5 = 5. Thus, \det(A_z) = 5 \cdot 1 + 6 \cdot (-2) + 3 \cdot 5 = 5 - 12 + 15 = 8, so z = 8/8 = 1. The solution is \mathbf{x} = (2, 1, 1).

Mathematical Justification

Determinant-Based Proof

The determinant-based proof of Cramer's rule begins with the linear system Ax = b, where A is an n \times n invertible matrix (so \det(A) \neq 0), x is the unknown column vector, and b is the given column vector of constants.^[10] Multiplying both sides of the equation by the adjugate matrix \adj(A) produces \adj(A) A x = \adj(A) b. A fundamental property of square matrices states that \adj(A) A = \det(A) I, where I is the n \times n identity matrix; this identity follows from the definition of the adjugate as the transpose of the cofactor matrix and the multilinearity of the determinant.^[11] Substituting this property yields \det(A) x = \adj(A) b, and dividing by \det(A) (valid since it is nonzero) gives the solution x = \frac{\adj(A) b}{\det(A)}.^[10] To derive the component-wise form, consider the i-th entry of this solution: x_i = \frac{[\adj(A) b]_i}{\det(A)}. The i-th row of \adj(A) consists of the cofactors C_{j i} for j = 1, \dots, n, since \adj(A) is the transpose of the cofactor matrix C (where C_{j i} = (-1)^{j+i} \det(M_{j i}) and M_{j i} is the minor obtained by deleting row j and column i from A). Thus,

[\adj(A) b]_i = \sum_{j=1}^n C_{j i} b_j. $$ This expression is the cofactor expansion of the determinant of the matrix $ A_i $, which is $ A $ with its $ i $-th column replaced by $ b $; multilinearity of the determinant along the $ i $-th column implies $ \det(A_i) = \sum_{j=1}^n C_{j i} b_j $.[](https://mathresearch.utsa.edu/wiki/index.php?title=Cramer%27s_Rule)[](https://www.math.purdue.edu/~shao92/documents/Proof_Cramer_rule.pdf) Therefore, $ x_i = \frac{\det(A_i)}{\det(A)} $, establishing Cramer's rule.[](http://scipp.ucsc.edu/~haber/ph116A/adjugate.pdf) This proof hinges on the Leibniz formula for the [determinant](/page/Determinant), which expresses $ \det(A) $ as a sum over permutations and underpins the alternating and multilinear properties used in cofactor expansions and the adjugate construction. ### Geometric Interpretation Cramer's rule gains an intuitive geometric meaning through the lens of vector spaces, where the [determinant](/page/Determinant) of a square matrix measures the signed [volume](/page/Volume) of the [parallelepiped](/page/Parallelepiped) formed by its column vectors. For an [invertible matrix](/page/Invertible_matrix) $A$ with columns $\mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n$ in $\mathbb{R}^n$, the value $\det(A)$ represents this signed [volume](/page/Volume), capturing both magnitude (absolute [volume](/page/Volume)) and [orientation](/page/Orientation) (positive or negative [sign](/page/Sign) depending on whether the columns form a right-handed or left-handed basis).[](https://math.libretexts.org/Courses/Irvine_Valley_College/Math_26%253A_Introduction_to_Linear_Algebra/02%253A_Linear_Transformations_and_Matrix_Algebra/2.07%253A_Determinants_and_Volumes/2.7.01%253A_Determinants_for_Area_and_Volume) This [volume](/page/Volume) interpretation underscores how the linear transformation defined by $A$ scales the unit [hypercube](/page/Hypercube), providing a foundational geometric role for determinants in solving systems $A\mathbf{x} = \mathbf{b}$.[](https://ntrs.nasa.gov/api/citations/20210022778/downloads/CramerCross_accepted.pdf) In the context of Cramer's rule, consider the matrix $A_i$ obtained by replacing the $i$-th column of $A$ with the right-hand side vector $\mathbf{b}$. The determinant $\det(A_i)$ then gives the signed volume of the parallelepiped spanned by the columns of $A_i$, which effectively measures the "height" or projection-related contribution of $\mathbf{b}$ in the direction orthogonal to the hyperplane formed by the other columns $\mathbf{a}_1, \dots, \mathbf{a}_{i-1}, \mathbf{a}_{i+1}, \dots, \mathbf{a}_n$.[](https://ntrs.nasa.gov/api/citations/20210022778/downloads/CramerCross_accepted.pdf) Geometrically, this replacement interprets $\mathbf{b}$ as taking the place of $\mathbf{a}_i$, yielding a volume that reflects how $\mathbf{b}$ fits into the span of the columns when adjusted along the $i$-th direction. The solution component $ x_i $ is the ratio of these signed volumes: $ x_i = \frac{\det(A_i)}{\det(A)} $, indicating the signed scaling factor needed along $\mathbf{a}_i$ to express $\mathbf{b}$ in the basis defined by the columns of $A$.[](https://www.datacamp.com/tutorial/cramers-rule) This volume ratio provides a direct visualization in lower dimensions. In two dimensions, $\det(A)$ is the signed area of the parallelogram formed by $\mathbf{a}_1$ and $\mathbf{a}_2$, while $\det(A_1)$ (replacing $\mathbf{a}_1$ with $\mathbf{b}$) is the signed area of the parallelogram spanned by $\mathbf{b}$ and $\mathbf{a}_2$; thus, $x_1$ is the ratio of these areas, representing the relative "stretch" along $\mathbf{a}_1$ to reach $\mathbf{b}$.[](https://ntrs.nasa.gov/api/citations/20210022778/downloads/CramerCross_accepted.pdf) Similarly, in three dimensions, $\det(A)$ denotes the signed volume of the parallelepiped defined by $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$, and replacing one column with $\mathbf{b}$ adjusts this volume to reflect the coordinate $x_i$ as a volume quotient, illustrating how $\mathbf{b}$ decomposes into components parallel to each basis vector through these scaled volumes.[](https://math.libretexts.org/Courses/Irvine_Valley_College/Math_26%253A_Introduction_to_Linear_Algebra/02%253A_Linear_Transformations_and_Matrix_Algebra/2.07%253A_Determinants_and_Volumes/2.7.01%253A_Determinants_for_Area_and_Volume) This perspective highlights Cramer's rule not merely as an algebraic tool but as a manifestation of spatial decomposition in vector spaces. ## Extensions and Special Cases ### Matrix Inversion via Cramer's Rule To compute the [inverse](/page/Inverse) of an invertible [square matrix](/page/Square_matrix) $A$, solve the [linear system](/page/Linear_system) $A \mathbf{x} = \mathbf{e}_j$ for each [standard basis](/page/Standard_basis) vector $\mathbf{e}_j$ ($j = 1, \dots, n$), where $\mathbf{e}_j$ is the vector with a 1 in the $j$-th position and zeros elsewhere; the resulting solution $\mathbf{x}$ forms the $j$-th column of $A^{-1}$.[](https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/f6e46da0d783d8f9c0a25c407c76166a_MIT18_06SCF11_Ses2.7sum.pdf) Applying Cramer's rule to this system, the $i$-th entry of $\mathbf{x}$, denoted $(A^{-1})_{ij}$, is given by $\frac{\det(A^{(i,j)})}{\det(A)}$, where $A^{(i,j)}$ is the matrix obtained by replacing the $i$-th column of $A$ with $\mathbf{e}_j$. The determinant $\det(A^{(i,j)})$ equals the cofactor $C_{ji}$ of the entry $a_{ji}$ in $A$, which is $(-1)^{j+i}$ times the determinant of the submatrix obtained by deleting row $j$ and column $i$ from $A$.[](https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/f6e46da0d783d8f9c0a25c407c76166a_MIT18_06SCF11_Ses2.7sum.pdf) Thus, $(A^{-1})_{ij} = \frac{C_{ji}}{\det(A)}$. The matrix whose $(i,j)$-th entry is $C_{ji}$ is the adjugate of $A$, denoted $\adj(A)$, which is the transpose of the cofactor matrix of $A$. The full inverse is therefore expressed as

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

[](https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/f6e46da0d783d8f9c0a25c407c76166a_MIT18_06SCF11_Ses2.7sum.pdf) ### Singular and Inconsistent Systems When the determinant of the [coefficient matrix](/page/Coefficient_matrix) $A$ in a [linear system](/page/Linear_system) $Ax = b$ is zero, that is, $\det(A) = 0$, Cramer's rule becomes [undefined](/page/Undefined) because it requires division by $\det(A)$, resulting in indeterminate forms such as $0/0$.[](https://mathresearch.utsa.edu/wiki/index.php?title=Cramer%27s_Rule) In this case, the system may have either no [solution](/page/Solution) or infinitely many solutions, and further analysis is needed to determine [consistency](/page/Consistency).[](https://openlab.citytech.cuny.edu/?get_group_doc=20321/1493249485-cramer_rules_2x2_and_3x3.pdf) To assess the situation using elements of Cramer's rule, one examines the determinants of the matrices $A_i$ obtained by replacing the $i$-th column of $A$ with the constant vector $b$. If $\det(A) = 0$ but $\det(A_i) \neq 0$ for at least one $i$, the system is inconsistent and has no solution. If $\det(A) = 0$ and $\det(A_i) = 0$ for all $i$, the system may be consistent with infinitely many solutions or may be inconsistent; further analysis, such as comparing the ranks of $A$ and the [augmented matrix](/page/Augmented_matrix) $[A|b]$, is required to determine consistency.[](https://mathresearch.utsa.edu/wiki/index.php?title=Cramer%27s_Rule) A more general condition for consistency of the system is that the rank of $A$ equals the rank of the [augmented matrix](/page/Augmented_matrix) $[A|b]$.[](https://math.colorado.edu/~nita/SystemsofLinearEquations.pdf) For the special case of a homogeneous [system](/page/System) $Ax = 0$, where $b = 0$, the trivial [solution](/page/Solution) $x = 0$ always exists. This trivial solution is unique [if and only if](/page/If_and_only_if) $\det(A) \neq 0$; otherwise, if $\det(A) = 0$, there are infinitely many nontrivial solutions.[](https://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/system/system.html) The [determinant](/page/Determinant) thus acts as a test for the invertibility of $A$, confirming a unique [solution](/page/Solution) in the invertible case.[](https://www.math.umd.edu/~immortal/MATH246/lecturenotes/ch2-3.pdf) ## Applications ### Explicit Solutions for Small Systems Cramer's rule provides closed-form expressions for the solutions of small linear systems by expressing each [variable](/page/Variable) as the [ratio](/page/Ratio) of two [determinant](/page/Determinant)s. For a 2×2 system of the form

\begin{cases} a x + b y = e \ c x + d y = f \end{cases}

where the [coefficient matrix](/page/Coefficient_matrix) has nonzero [determinant](/page/Determinant) $\Delta = ad - bc$, the solutions are given by

x = \frac{\det\begin{pmatrix} e & b \ f & d \end{pmatrix}}{\Delta} = \frac{df - be}{ad - bc},

y = \frac{\det\begin{pmatrix} a & e \ c & f \end{pmatrix}}{\Delta} = \frac{ae - cd}{ad - bc}.

These formulas arise directly from substituting the right-hand side vector into the respective columns of the coefficient matrix and computing the resulting determinants.[22] For a 3\times3 system

\begin{cases} a_{11} x + a_{12} y + a_{13} z = b_1 \ a_{21} x + a_{22} y + a_{23} z = b_2 \ a_{31} x + a_{32} y + a_{33} z = b_3 \end{cases}

with coefficient matrix $A$ having nonzero determinant $\Delta = \det(A)$, the solutions are $x = \det(A_x)/\Delta$, $y = \det(A_y)/\Delta$, and $z = \det(A_z)/\Delta$, where $A_x$, $A_y$, and $A_z$ are formed by replacing the first, second, and third columns of $A$ with the column vector $(b_1, b_2, b_3)^T$, respectively.[](https://mathresearch.utsa.edu/wiki/index.php?title=Cramer%27s_Rule) The determinant $\Delta$ expands via cofactor expansion along the first row as

\Delta = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}).

Similarly,

\det(A_x) = b_1(a_{22}a_{33} - a_{23}a_{32}) - b_2(a_{12}a_{33} - a_{13}a_{32}) + b_3(a_{12}a_{23} - a_{13}a_{22}),

\det(A_y) = a_{11}(b_2 a_{33} - b_3 a_{23}) - a_{21}(b_1 a_{33} - b_3 a_{13}) + a_{31}(b_1 a_{23} - b_2 a_{13}),

\det(A_z) = a_{11}(a_{22} b_3 - a_{23} b_2) - a_{12}(a_{21} b_3 - a_{23} b_1) + a_{13}(a_{21} b_2 - a_{22} b_1).

These expansions follow from the standard Laplace expansion of the 3×3 determinants.[](https://www.math.utah.edu/~gustafso/determinants.pdf) ### Uses in Differential Geometry and ODEs In differential geometry, Cramer's rule facilitates the computation of Christoffel symbols within the Ricci calculus framework, where these symbols appear as solutions to linear systems derived from the metric tensor and its partial derivatives in local coordinates. For a surface parametrized by coordinates $u$ and $v$, the second partial derivatives of the position vector, such as $\mathbf{S}_{uu}$, are expressed as linear combinations $\Gamma^1_{11} \mathbf{S}_u + \Gamma^2_{11} \mathbf{S}_v + d_1 \mathbf{N}$, leading to a system of equations from inner products with the basis vectors: $\Gamma^1_{11} E + \Gamma^2_{11} F = \frac{1}{2} E_u$ and $\Gamma^1_{11} F + \Gamma^2_{11} G = F_u - \frac{1}{2} E_v$, where $E = \langle \mathbf{S}_u, \mathbf{S}_u \rangle$, $F = \langle \mathbf{S}_u, \mathbf{S}_v \rangle$, and $G = \langle \mathbf{S}_v, \mathbf{S}_v \rangle$ are coefficients of the first fundamental form. Applying Cramer's rule to this 2×2 system yields explicit formulas for $\Gamma^1_{11}$ and $\Gamma^2_{11}$ in terms of $E, F, G,$ and their partials, with analogous procedures for other indices.[](https://www.math.purdue.edu/~arapura/preprints/diffgeom9.pdf) This approach extends to higher dimensions in Riemannian manifolds, where Christoffel symbols of the second kind $\Gamma^k_{ij}$ solve larger linear systems involving the inverse metric $g^{kl}$ and Christoffel symbols of the first kind, ensuring coordinate-invariant expressions for covariant derivatives and geodesic equations.[](https://www.math.purdue.edu/~arapura/preprints/diffgeom9.pdf) In the context of implicit differentiation for multivariable functions, Cramer's rule provides a systematic way to compute partial derivatives when variables are related by equations like $F(x, y, z) = 0$. Differentiating yields the total differential $F_x \, dx + F_y \, dy + F_z \, dz = 0$, which forms a linear system in the differentials; solving for, say, $\partial y / \partial x$ (treating $z$ as constant) involves Cramer's rule on the subsystem excluding $dz$. For multiple implicit equations $F_i(x_1, \dots, x_n, y_1, \dots, y_m) = 0$, the Jacobian matrix of partials with respect to the $y_j$ allows Cramer's rule to express $\partial y_k / \partial x_i$ as ratios of determinants, aligning with the implicit function theorem's conditions for local solvability.[](https://jhqian.org/econ2403/mathbackground.pdf) A specific application arises in [2D](/page/2D) implicit curves defined by $F(x, y) = 0$, where the [derivative](/page/Derivative) $dy/dx$ satisfies the [linear system](/page/Linear_system) $F_x \, dx + F_y \, dy = 0$. Using Cramer's rule on the [coefficient matrix](/page/Coefficient_matrix) $\begin{pmatrix} F_x & F_y \\ 1 & 0 \end{pmatrix}$ (with right-hand side $(0, 1)^T$ for normalized $dx = 1$) gives

\frac{dy}{dx} = \frac{\det \begin{pmatrix} F_x & 0 \ 1 & 1 \end{pmatrix}}{\det \begin{pmatrix} F_x & F_y \ 1 & 0 \end{pmatrix}} = -\frac{F_x}{F_y},

assuming $F_y \neq 0$, which matches the standard formula and highlights the rule's role in geometric interpretations of tangents.[](https://math.stackexchange.com/questions/1157921/solving-for-a-partial-derivative-with-cramers-rule) For systems of ordinary differential equations (ODEs), Cramer's rule aids in solving constant-coefficient nonhomogeneous [linear system](/page/Linear_system)s $\mathbf{y}' = A \mathbf{y} + \mathbf{g}(t)$, particularly via [variation of parameters](/page/Variation_of_parameters). After finding the fundamental matrix $\Phi(t)$ from the homogeneous solution (using eigenvalues of $A$), the particular solution assumes $\mathbf{y}_p = \Phi(t) \mathbf{u}(t)$, leading to $\mathbf{u}'(t) = \Phi^{-1}(t) \mathbf{g}(t)$. For small dimensions like $n=2$, Cramer's rule explicitly solves this [linear system](/page/Linear_system) for the components of $\mathbf{u}'$, yielding determinants involving entries of $\Phi^{-1}$ and $\mathbf{g}$. This method is especially useful for explicit computation in low-order systems, such as coupled oscillators or predator-prey models.[](https://courses.lumenlearning.com/calculus3/chapter/nonhomogeneous-linear-equations/) ### Applications in Engineering and Economics In electrical engineering, Cramer's rule is applied in circuit analysis, particularly nodal voltage and mesh current methods, to solve systems of linear equations for unknown voltages or currents. For example, in nodal analysis, Kirchhoff's current law yields equations at each node, solved using determinants to find node potentials efficiently for small networks.[](https://en.wikiversity.org/wiki/Electric_Circuit_Analysis/Nodal_Analysis) In economics, the rule is used in linear models such as Leontief input-output systems to determine production levels or in market equilibrium models for finding prices and quantities. It also supports comparative statics analysis, evaluating how changes in exogenous variables affect endogenous ones through determinant ratios.[](https://huwdixon.org/teaching/MScMaths/L4.pdf) ## Alternative Derivations ### Linear Algebra Perspective In the context of linear algebra, Cramer's rule provides an explicit formula for the coordinates of a [vector](/page/Vector) with respect to a given basis formed by the columns of the [coefficient matrix](/page/Coefficient_matrix). Consider the [linear system](/page/Linear_system) $Ax = b$, where $A$ is an $n \times n$ [invertible matrix](/page/Invertible_matrix) with columns $\mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n$, which constitute a basis for the [vector space](/page/Vector_space) $\mathbb{R}^n$ since $\det(A) \neq 0$. The unique solution $\mathbf{x} = (x_1, x_2, \dots, x_n)^T$ satisfies $b = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \dots + x_n \mathbf{a}_n$, where the components $x_i$ are precisely the coordinates of $b$ in this basis. This representation underscores the fundamental role of bases in solving [linear system](/page/Linear_system)s abstractly.[](https://users.math.msu.edu/users/parker/309/Lang-linear-algebra.pdf) To derive the explicit form of these coordinates using determinant properties, note that the [determinant](/page/Determinant) is a multilinear [function](/page/Function) of the columns and vanishes when any two columns are identical. Let $A^{(i)}$ denote the matrix obtained by replacing the $i$-th column of $A$ with $b$. By multilinearity in the $i$-th column position,

\det(A^{(i)}) = \det(\mathbf{a}1, \dots, \mathbf{a}{i-1}, b, \mathbf{a}_{i+1}, \dots, \mathbf{a}n) = \sum{k=1}^n x_k \det(\mathbf{a}1, \dots, \mathbf{a}{i-1}, \mathbf{a}k, \mathbf{a}{i+1}, \dots, \mathbf{a}_n).

For each term where $k \neq i$, the resulting matrix has identical columns in the $i$-th and $k$-th positions (both $\mathbf{a}_k$), so the determinant is zero. The only surviving term is when $k = i$, yielding $\det(A^{(i)}) = x_i \det(A)$. Thus, $x_i = \det(A^{(i)}) / \det(A)$, establishing Cramer's rule as the coordinate extraction formula in this basis.[](https://users.math.msu.edu/users/parker/309/Lang-linear-algebra.pdf) This perspective highlights Cramer's rule as the coordinate expression of the unique linear combination theorem in finite-dimensional vector spaces, relying solely on the algebraic structure of bases and the characterizing properties of the determinant function, without recourse to matrix inverses or cofactor expansions.[](https://users.math.msu.edu/users/parker/309/Lang-linear-algebra.pdf) ### Geometric Algebra Approach In geometric algebra (GA), also known as Clifford algebra, vectors and higher-grade elements (multivectors) provide a natural framework for deriving Cramer's rule through the outer product operation, which encodes oriented volumes and linear dependence. Consider a system of linear equations $ \sum_{i=1}^n x_i \mathbf{a}_i = \mathbf{b} $, where $\mathbf{a}_1, \dots, \mathbf{a}_n, \mathbf{b} \in \mathbb{R}^n$ are vectors forming the columns of the coefficient matrix $A$ and the right-hand side vector, respectively. The outer product $\mathbf{a}_1 \wedge \mathbf{a}_2 \wedge \dots \wedge \mathbf{a}_n$ is an $n$-blade (pseudoscalar multiple) whose scalar coefficient, extracted as $\langle (\mathbf{a}_1 \wedge \dots \wedge \mathbf{a}_n) I^{-1} \rangle$ with $I = \mathbf{e}_1 \wedge \dots \wedge \mathbf{e}_n$ the unit pseudoscalar, equals $\det(A)$. This scalar measures the signed volume of the parallelepiped spanned by the $\mathbf{a}_i$, and a nonzero determinant ensures linear independence.[](https://math.mit.edu/~dunkel/Teach/18.S996_2022S/books/Hestenes-Sobczyk1984_Book_CliffordAlgebraToGeometricCalc.pdf) To isolate the unknown coefficient $x_k$, form the $(n-1)$-blade $B_k = \mathbf{a}_1 \wedge \dots \wedge \mathbf{a}_{k-1} \wedge \mathbf{a}_{k+1} \wedge \dots \wedge \mathbf{a}_n$. Outer multiply the original equation on both sides by $B_k$:

\sum_{i=1}^n x_i (\mathbf{a}_i \wedge B_k) = \mathbf{b} \wedge B_k.

The left-hand side simplifies due to the antisymmetry of the outer product: for $i \neq k$, $\mathbf{a}_i \wedge B_k = 0$ because $B_k$ already includes $\mathbf{a}_i$, making the product vanish (repeated factors yield zero). For $i = k$, $\mathbf{a}_k \wedge B_k = (-1)^{k-1} (\mathbf{a}_1 \wedge \dots \wedge \mathbf{a}_n)$, where the sign arises from permuting $\mathbf{a}_k$ to its position. Thus, the equation reduces to

x_k (-1)^{k-1} (\mathbf{a}_1 \wedge \dots \wedge \mathbf{a}_n) = \mathbf{b} \wedge B_k.

Projecting onto the scalar part via multiplication by $I^{-1}$ and taking the scalar coefficient yields

x_k \det(A) = (-1)^{k-1} \langle (\mathbf{b} \wedge B_k) I^{-1} \rangle,

where $\langle (\mathbf{b} \wedge B_k) I^{-1} \rangle = \det(A_k)$ is the determinant of the matrix obtained by replacing the $k$-th column of $A$ with $\mathbf{b}$. Solving for $x_k$ gives the Cramer's rule formula $x_k = \det(A_k) / \det(A)$, with the sign absorbed into the conventional definition of determinants via column permutations. This derivation highlights linear dependence through grade annihilation in the outer product and extends naturally to the reciprocal frame, where basis duals $\mathbf{a}^k = (-1)^{k-1} B_k (\mathbf{a}_1 \wedge \dots \wedge \mathbf{a}_n)^{-1}$ satisfy $\mathbf{a}^k \cdot \mathbf{a}_i = \delta_{ki}$, allowing $x_i = \mathbf{b} \cdot \mathbf{a}^i$.[](https://math.mit.edu/~dunkel/Teach/18.S996_2022S/books/Hestenes-Sobczyk1984_Book_CliffordAlgebraToGeometricCalc.pdf)[](https://pages.astro.umd.edu/~jpha/GAandGC.pdf) For small systems, such as $n=2$ in $\mathbb{R}^2$, let $\mathbf{a}_1 = a_{11} \mathbf{e}_1 + a_{21} \mathbf{e}_2$, $\mathbf{a}_2 = a_{12} \mathbf{e}_1 + a_{22} \mathbf{e}_2$, $\mathbf{b} = b_1 \mathbf{e}_1 + b_2 \mathbf{e}_2$. The [pseudoscalar](/page/Pseudoscalar) is $\mathbf{a}_1 [\wedge](/page/Wedge) \mathbf{a}_2 = (a_{11} a_{22} - a_{21} a_{12}) \mathbf{e}_1 \wedge \mathbf{e}_2$, so $\det(A) = a_{11} a_{22} - a_{21} a_{12}$. For $x_1$, $B_1 = \mathbf{a}_2$, yielding $\mathbf{b} \wedge \mathbf{a}_2 = (b_1 a_{22} - b_2 a_{12}) \mathbf{e}_1 \wedge \mathbf{e}_2$, and $x_1 = (b_1 a_{22} - b_2 a_{12}) / \det(A)$, matching Cramer's rule. This GA formulation unifies the algebraic solution with geometric volume ratios, emphasizing the role of multivectors in capturing dependencies without explicit matrix inverses.[](https://math.mit.edu/~dunkel/Teach/18.S996_2022S/books/Hestenes-Sobczyk1984_Book_CliffordAlgebraToGeometricCalc.pdf)

References

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Cramer's Rule - an overview | ScienceDirect Topics
Cramer's rule is defined as a method for directly computing the solution to a system of linear equations, using determinants, provided one exists.
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Three methods for solving systems of linear equations - IOP Science
However, Cramer's rule needs to be under the condition of square matrices. At the same time, Gaussian elimination requires three elementary row operations, and.
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Feb 28, 2019 · In this study, Cramer's rule was considered an appropriate method to enhance PMTs' understanding of the solution of a system of equation, since ...
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Gabriel Cramer (1704 - 1752) - Biography - MacTutor
Gabriel Cramer worked on analysis and determinants. He is best known for his formula for solving simultaneous equations. Thumbnail of Gabriel Cramer View two ...
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Matrices and determinants - MacTutor History of Mathematics
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Mathematical Treasure: Cramer on His Rule and His Paradox
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### Summary of Cramer's Rule
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Cramer's Rule -- from Wolfram MathWorld
Cramer's Rule ; {a_1x+b_1y+c_1z=d_1; a_2x+b_2y+c_2z=d_2. (1) ; D=|a_1 b_1 c_1; a_2 b_2 c_2; a_3 b_3 c_3|. (2) ; x|a_1 b_1 c_1; a_2 b_2 c_2; a_3 b_3 c_3|=|. (3) ...
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Solving for a Partial Derivative with Cramer's Rule
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Nonhomogeneous Linear Equations | Calculus III - Lumen Learning
Cramer's rule would say that u prime is going to be equal to a fraction, and the numerator is the determinant of a matrix, and the denominator is a determinant ...
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CRAMER'S RULE. The properties of the preceding section can be used to prove a well- known rule used in solving linear equations. Theorem 4.1 ( Cramer's rule).<|control11|><|separator|>
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Since Geometric Algebra smoothly integrates quatemions with the conventional vector algebra, it makes the full power of both systems available together for ...
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Feb 13, 2014 · Cramer's rule. Problem: In R4 solve v = c1u1 + c2u2 + c3u3 + c4u4 for, say, c2. Solution: Outer multiply the equation by u1 on the left and ...<|control11|><|separator|>