General & First-year
Linear Algebra Lab
A matrix is just a recipe for where space goes. Drag the four entries and the whole grid shears, rotates, stretches and flips: the basis vectors î → (a, c) and ĵ → (b, d) move, the unit square becomes a parallelogram whose area is the determinant (negative means orientation flipped), and the dashed lines are the eigenvector directions — the special lines the transform only stretches and never rotates. When the matrix is a pure rotation there are no real eigenvectors, and the readout says so.
0.75
Determinant
2.00
Trace
1.50
λ₁
0.50
λ₂
The method, with your numbers
Determinant, trace, eigenvalues
- 1
Read off where the basis vectors land
î → (a, c) ĵ → (b, d)
î → (1, 0.5) ĵ → (0.5, 1)
The columns of the matrix ARE the landed basis vectors — that's how to read any matrix.
- 2
Determinant — the area scale factor
det = a·d − b·c
det = 1·1 − 0.5·0.5 = 0.75
Areas scale by |det| = 0.75× — watch the shaded parallelogram grow or shrink.
- 3
Trace — sum of the diagonal
tr = a + d
tr = 1 + 1 = 2
Quick self-check for later: the trace always equals λ₁ + λ₂.
- 4
Discriminant of the characteristic equation
Δ = tr² − 4·det
Δ = 2² − 4·0.75 = 1
Δ ≥ 0 means real eigenvalues (directions that only stretch); Δ < 0 means the map rotates every vector.
- 5
Eigenvalues — the pure stretch factors
λ₁,₂ = (tr ± √Δ) / 2
λ₁ = (2 + 1) / 2 = 1.5 λ₂ = (2 − 1) / 2 = 0.5
The dashed lines are the eigenvector directions — vectors on them only stretch by λ, never rotate. Check: λ₁·λ₂ = det.
Drag any control above — every number here recalculates. Want this method for any problem? Step Sheets →
The matrix is just where the basis vectors land: î → (a, c), ĵ → (b, d). The determinant is how much area is stretched (negative = flipped); the dashed lines are eigenvector directions — the ones the transform only stretches, never rotates.
How to use this simulation
A matrix is just a recipe for where space goes. Drag the four entries and the whole grid shears, rotates, stretches and flips: the basis vectors î → (a, c) and ĵ → (b, d) move, the unit square becomes a parallelogram whose area is the determinant (negative means orientation flipped), and the dashed lines are the eigenvector directions — the special lines the transform only stretches and never rotates. When the matrix is a pure rotation there are no real eigenvectors, and the readout says so.
Everything runs in your browser — no sign-up, no download. Change a value and the result updates instantly, so you can build a feel for how each input shapes the outcome. It pairs with Crameleon's practice exams and step sheets when you want to go from intuition to working the problems.