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Algebra
Matrix Operations Cheat Sheet
Complete matrix operations cheat sheet with addition, multiplication, determinants, inverses, eigenvalues, and transformations. Free PDF download for linear algebra.
Basic Operations
| Name | Formula | Notes |
|---|---|---|
| Addition | Add corresponding elements | |
| Scalar Multiplication | Multiply each element by k | |
| Transpose | Swap rows and columns |
Matrix Multiplication
| Name | Formula | Notes |
|---|---|---|
| Definition | Row × Column dot product | |
| 2×2 Example | ||
| Not Commutative | Order matters! | |
| Associative | ||
| Distributive |
2×2 Determinant & Inverse
| Name | Formula | Notes |
|---|---|---|
| 2×2 Determinant | ||
| 2×2 Inverse | Only exists if det ≠ 0 | |
| Inverse Property | Identity matrix result |
3×3 Determinant
| Name | Formula | Notes |
|---|---|---|
| Rule of Sarrus | For 3×3 matrices | |
| Cofactor Expansion | Expand along any row/column |
Special Matrices
| Name | Formula | Notes |
|---|---|---|
| Identity Matrix | AI = IA = A | |
| Zero Matrix | A + O = A | |
| Symmetric | Equal to its transpose | |
| Orthogonal | Transpose equals inverse |
Eigenvalues & Eigenvectors
| Name | Formula | Notes |
|---|---|---|
| Definition | v is eigenvector, λ is eigenvalue | |
| Characteristic Equation | Solve for eigenvalues | |
| 2×2 Eigenvalues | tr(A) = a + d (trace) |
Determinant Properties
| Name | Formula | Notes |
|---|---|---|
| Product | ||
| Inverse | ||
| Transpose | ||
| Scalar Multiple | n is matrix size |
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