Matrix Inverse, Determinant & Rank

Compute the determinant, rank, nullity, and inverse of a square matrix using Gaussian elimination with partial pivoting. Intermediate matrices are shown at each major step.

⊞ Input Matrix

Matrix size (n × n)

Or paste matrix (space/comma separated rows, newline between)

✓ Results

Enter a matrix and click Solve.

Technical Description & Theory

What is Matrix Inverse Determinant Rank Calculator and Why it Matters?

This calculator computes the determinant, rank, nullity, and inverse of a square matrix. These quantities are fundamental in linear algebra for solving systems, analyzing linear transformations, and checking invertibility. Engineering and research students use them in structural analysis, control theory, and numerical methods.

Mathematical Formula and Theory

For an n×n matrix A, the determinant det(A) is computed via Gaussian elimination with partial pivoting. Rank is the number of non-zero pivots. The inverse exists only when det(A) ≠ 0 and satisfies A A⁻¹ = I. Nullity follows from the rank-nullity theorem: rank(A) + nullity(A) = n.

How to Use This Calculator - Step by Step

Select matrix size (2×2 to 5×5).
Enter matrix entries or paste space-separated rows.
Click Solve to obtain determinant, rank, nullity, and inverse (if it exists).
Review the step notes describing the underlying elimination process.

Solved Example with Full Calculation

Consider A = [[2,1,1],[1,3,2],[1,0,0]]. Using Gaussian elimination the pivots are non-zero. det(A) = −1, rank = 3, nullity = 0. The inverse is [[0,0,1],[−2,1,1],[3,−1,−1]]. Since rank equals n, A is invertible and full rank.