Linear Algebra Tool

📐 Linear Algebra Tool matrices · vectors · eigenvalues

🧮 matrix operations · solve systems · determinants · inverses · eigenvalues · more

➕ Matrix Operations

Matrix A (rows separated by ;, numbers by ,) 1,2,3;4,5,6;7,8,10 Matrix B (rows separated by ;, numbers by ,) 10,8,7;6,5,4;3,2,1 Operation Addition (A + B) Subtraction (A - B) Multiplication (A × B) Transpose (Aᵀ)

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Enter matrices in the format: row1; row2; row3

🧮 Determinant

Matrix (rows separated by ;, numbers by ,) 1,2,3;4,5,6;7,8,10

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det(A) · matrix must be square

🔢 Inverse

Matrix (rows separated by ;, numbers by ,) 1,2,3;4,5,6;7,8,10

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A⁻¹ · matrix must be invertible (det ≠ 0)

📊 Solve Linear System

Matrix A (coefficients) 1,2,3;4,5,6;7,8,10 Vector b (constants)

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Uses Gaussian elimination · returns x

📈 Eigenvalues

Matrix (rows separated by ;, numbers by ,) 4,1,2;1,3,1;2,1,4

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Eigenvalues (λ) · characteristic polynomial

📏 Vector Operations

Vector u (comma separated) Vector v (comma separated) Operation Dot Product (u·v) Cross Product (u×v) Norm (|u|) Angle between u and v

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Vectors in R³ · dot, cross, norm, angle

📊 Matrix Rank

Matrix (rows separated by ;, numbers by ,) 1,2,3;4,5,6;7,8,10

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Rank = number of linearly independent rows/columns

📊 Trace

Matrix (rows separated by ;, numbers by ,) 1,2,3;4,5,6;7,8,10

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tr(A) = sum of diagonal elements

📊 Adjugate Matrix

Matrix (rows separated by ;, numbers by ,) 1,2,3;4,5,6;7,8,10

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adj(A) = transpose of cofactor matrix

📐 Gram-Schmidt

Vectors (rows separated by ;, numbers by ,) 1,1,0;0,1,1;1,0,1

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Orthonormal basis from given vectors

📐 Linear Algebra Tool – 10-in-1
This comprehensive tool provides essential linear algebra functions:

1. Matrix Operations – add, subtract, multiply, transpose.
2. Determinant – compute det(A) for square matrices.
3. Inverse – find A⁻¹ for invertible matrices.
4. Solve Linear System – solve Ax = b using Gaussian elimination.
5. Eigenvalues – compute eigenvalues and characteristic polynomial.
6. Vector Operations – dot product, cross product, norm, angle.
7. Rank – determine matrix rank.
8. Trace – sum of diagonal elements.
9. Adjugate – compute the adjugate matrix.
10. Gram-Schmidt – orthonormalize a set of vectors.

💡 Tips:
• Enter matrices as: row1; row2; row3 (e.g., 1,2,3;4,5,6;7,8,9)
• Enter vectors as: 1,2,3 (comma separated)
• All calculations are performed client-side – no data is sent anywhere.
• Perfect for students, engineers, and data scientists.

📖 Linear Algebra Tool – detailed explanation

➕ Matrix Operations
• Addition: (A+B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ
• Subtraction: (A-B)ᵢⱼ = Aᵢⱼ - Bᵢⱼ
• Multiplication: (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ
• Transpose: (Aᵀ)ᵢⱼ = Aⱼᵢ

🧮 Determinant
The determinant is a scalar value that indicates whether a matrix is invertible. det(A) ≠ 0 means the matrix is invertible.

🔢 Inverse
A⁻¹ is the matrix such that A·A⁻¹ = I. Only square matrices with det(A) ≠ 0 have an inverse.

📊 Solve Linear System
Solves Ax = b using Gaussian elimination with back-substitution. Returns the solution vector x.

📈 Eigenvalues
Eigenvalues λ satisfy det(A - λI) = 0. They are used in stability analysis, PCA, and quantum mechanics.

📏 Vector Operations
• Dot product: u·v = Σ uᵢvᵢ
• Cross product: u×v = (u₂v₃-u₃v₂, u₃v₁-u₁v₃, u₁v₂-u₂v₁)
• Norm: |u| = √(Σ uᵢ²)
• Angle: θ = arccos((u·v)/(|u||v|))

📊 Rank
The rank is the number of linearly independent rows or columns. It determines the dimension of the column space.

⚡ Educational tool · works offline · no server

✅ linear algebra tool · 10-in-1 · dynamic · no server