What does the determinant of a matrix tell you?

The determinant measures how a linear transformation scales areas (2D) or volumes (3D). If det(A) = 2, the transformation doubles all areas. A negative determinant reverses orientation. A determinant of zero means the matrix is singular — it collapses space into a lower dimension and has no inverse.

How are eigenvalues computed for larger matrices?

For 2×2, eigenvalues come from the quadratic formula on the characteristic polynomial. For 3×3, Cardano's method solves the cubic characteristic equation analytically. For 4×4, the Faddeev–LeVerrier algorithm extracts the characteristic polynomial coefficients, then the Durand–Kerner iterative method finds all four roots numerically.

When is a matrix invertible?

A matrix is invertible if and only if its determinant is not zero (equivalently, its rank equals its size). The inverse is computed as A⁻¹ = (1/det) · adj(A), where adj(A) is the adjugate (transpose of the cofactor matrix). This works for any square matrix — 2×2, 3×3, or 4×4.

What is the rank of a matrix?

The rank of a matrix is the number of linearly independent rows (or columns). It is computed via Gaussian elimination by counting non-zero pivot rows. A matrix with rank equal to its size is full rank and invertible. A rank-deficient matrix has a non-trivial null space.

Matrix Calculator

Compute determinant, inverse, eigenvalues, trace, and rank for 2×2, 3×3, and 4×4 matrices. The 2×2 view shows the geometric transformation of the unit square with animated basis vectors.

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Matrix	--
Determinant	--
Trace	--
Rank	--
Inverse	--
Eigenvalues	--
Eigenvectors	--

The Math Behind It

Matrix Operations

Determinant (2×2): det(A) = ad − bc
Determinant (n×n): Computed via cofactor expansion along the first row
Inverse: A¹ = (1/det) · adj(A) — exists only when det ≠ 0
Trace: tr(A) = ∑ a_ii — sum of diagonal entries
Rank: Number of non-zero rows after Gaussian elimination

Eigenvalues & Eigenvectors

Characteristic equation: det(A − λI) = 0
2×2: Quadratic formula — λ = (tr ± √(tr² − 4·det)) / 2
3×3: Cubic characteristic polynomial solved via Cardano’s method
4×4: Faddeev–LeVerrier algorithm + Durand–Kerner numerical root finding
Eigenvectors satisfy Av = λv — directions only scaled by the matrix

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