Linear Equations Formulas
Complete mathematical formulas for solving systems of linear equations using various methods and techniques
1General System Representation
Standard Form
Matrix Form
Where:
Coefficient Matrix (A)
Contains all coefficients of the variables:
- a₁₁, a₁₂, ..., a₁ₙ: coefficients of variables in equation 1
- a₂₁, a₂₂, ..., a₂ₙ: coefficients of variables in equation 2
- aₘ₁, aₘ₂, ..., aₘₙ: coefficients of variables in equation m
Variable Vector (x)
Contains all unknown variables we want to solve for:
- x₁: first unknown variable
- x₂: second unknown variable
- x₃: third unknown variable
- xₙ: nth unknown variable
Constants Vector (b)
Contains the constant values (right-hand side):
- b₁: constant value for equation 1
- b₂: constant value for equation 2
- b₃: constant value for equation 3
- bₘ: constant value for equation m
Example: 3×3 System
For the system of equations:
This can be written in matrix form as where:
Coefficient Matrix A:
| 2 | 3 | -1 |
| 1 | -2 | 4 |
| 3 | 1 | 2 |
Contains the coefficients of x₁, x₂, x₃
Variable Vector x:
The unknown variables we solve for
Constants Vector b:
The constant values from right side
Matrix Equation Verification:
When we multiply matrix A by vector x, we get vector b, which represents our original system of equations.
Augmented Matrix
2Cramer's Method
Prerequisites
- System must be square (m = n)
- Coefficient matrix must be non-singular (det(A) ≠ 0)
- System must have a unique solution
Cramer's Rule Formula
Where A₁ is the matrix obtained by replacing the i-th column of A with vector b.
2×2 System Example
Solution for x₁:
Solution for x₂:
3×3 System Formulas
3Gaussian Elimination (Row Echelon Method)
Elementary Row Operations
- Row Swapping: Rᵢ ↔ Rⱼ
- Row Scaling: kRᵢ → Rᵢ (k ≠ 0)
- Row Addition: Rᵢ + kRⱼ → Rᵢ
Row Echelon Form
A matrix is in row echelon form if:
- All nonzero rows are above any rows of all zeros
- Leading entry of each row is to the right of the leading entry in the row above it
- All entries below a leading entry are zeros
Back Substitution
For an upper triangular system:
Pivot Strategy
To minimize numerical errors:
4Matrix Inversion Method
Solution Formula
Where A⁻¹ is the inverse of the coefficient matrix A.
Matrix Inverse Formula
Where adj(A) is the adjugate (adjoint) matrix of A.
Adjugate Matrix
Where C is the cofactor matrix, and Cᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ
2×2 Matrix Inverse
3×3 Matrix Inverse
For matrix:
The inverse is:
Cofactor Calculation
Where Mᵢⱼ is the (i,j)-minor (determinant of the (n-1)×(n-1) submatrix).
5Determinant Formulas
2×2 Determinant
3×3 Determinant (Rule of Sarrus)
Cofactor Expansion
Expansion along row i or column j.
Properties of Determinants
- for n×n matrix
6Types of Solutions
Unique Solution
- • det(A) ≠ 0
- • rank(A) = rank([A|b]) = n
- • Lines intersect at one point
Infinite Solutions
- • det(A) = 0
- • rank(A) = rank([A|b]) < n
- • Dependent equations
No Solution
- • det(A) = 0
- • rank(A) < rank([A|b])
- • Inconsistent system
7Computational Complexity
| Method | Time Complexity | Space Complexity | Stability |
|---|---|---|---|
| Cramer's Method | Poor | ||
| Gaussian Elimination | Good with pivoting | ||
| Matrix Inversion | Moderate |