Computational Algebra & Matrices
Master matrix transformations, visualize vector mappings, detect eigenvalues, and step through Gaussian elimination.
Interactive Sandbox
Transformations & Row Reduction Solver
Study coordinate grid deformations using $2 \times 2$ matrices, or input a system of linear equations and watch a row reduction solver execute Gaussian Elimination step-by-step.
Matrix Presets
Matrix Coefficients A
1.0
0.0
0.0
1.0
Eigenvector Detected! λ = 1.00
Vector Mappings
Input Vector v: [1.00, 1.00]
Mapped Vector v': [1.00, 1.00]
Determinant det(A): 1.00
Trace Step Solver
Matrix Mathematical Operations
Review core linear algebra formulas, determinant boundaries, and systems classifications.
Matrix Operations
2D Linear Mappings:
Applying a matrix multiplication to coordinates maps vectors to transformed vectors:
v' = A·v = [a·x + b·y, c·x + d·y]T
Determinant and Areas:
For a $2 \times 2$ matrix, the determinant $\det(A) = ad - bc$ measures the scaling factor of areas. If $\det(A) = 0$, the transformation squashes the entire 2D plane down into a 1D line or 0D point.
Eigenvalue Condition:
Eigenvectors preserve collinear directions, scaling by eigenvalues:
A·v = λ·v.
System Stability & Classification
Gaussian Reduction States:
Elementary row operations yield three potential results:
1. Consistent - Unique Solution:
RREF reduces to Identity $I$. All variables are solved cleanly ($x=1, y=2, z=3$).
2. Consistent - Infinite Solutions:
Yields row of zeros $\begin{bmatrix} 0 & 0 & 0 & | & 0 \end{bmatrix}$. Free parameters exist, meaning coordinate planes intersect along lines or sheets.
3. Inconsistent - No Solution:
Yields row of contradictions $\begin{bmatrix} 0 & 0 & 0 & | & k \end{bmatrix}$ (where $k \neq 0$). Indicates coordinate planes are parallel and never meet.
Self Evaluation
Matrix Algebra Quiz
Assess your understanding of matrix sizes, multiplication constraints, determinants, and row-reductions.
Question 1 of 5
Score: 0/0
Correct Answer!
Focus Vocabulary
- Singular Matrix: A square matrix that lacks an inverse ($A^{-1}$), occurring precisely when $\det(A) = 0$.
- Eigenvector: A non-zero vector whose direction remains unchanged under a linear transformation.
- RREF: Reduced Row Echelon Form, the final state of Gaussian elimination where pivots equal 1 and pivot columns equal 0.