Matrix Calculator with Step-by-Step Row Reduction
Perform matrix operations and follow each elementary row operation in a complete Gauss-Jordan reduction walkthrough.
Matrix operations with visible working
Every row reduction is recorded operation by operation — row swaps, pivot scaling, and elimination — with the state of the matrix after each step and the pivot cell highlighted, so Gauss-Jordan elimination can be followed rather than just trusted.
- Single-matrix: determinant, inverse, transpose, trace, rank, adjugate, integer powers, and reduced row echelon form.
- Two-matrix: addition, subtraction, multiplication, and scalar multiplication.
- Linear systems Ax = b, including no-solution and infinite-solution cases with a null-space basis.
- Eigenvalues in closed form for 1×1, 2×2, and 3×3 matrices. Matrices up to 8×8.
Frequently asked questions
- How do you find the reduced row echelon form of a matrix?
- Work left to right. For each column, pick a non-zero pivot (swapping rows if needed), scale that row so the pivot becomes 1, then subtract multiples of it from every other row so the rest of the column is zero. Repeat on the next column. This calculator shows each of those elementary row operations and the resulting matrix.
- When does a matrix have no inverse?
- A square matrix is singular — it has no inverse — exactly when its determinant is zero, which is the same as saying its rank is less than its size, or that its rows and columns are linearly dependent. A non-square matrix never has a true inverse, only one-sided pseudo-inverses.
- What do eigenvalues tell you?
- An eigenvalue λ of A is a scalar for which some non-zero vector v satisfies A·v = λ·v: the matrix stretches that direction without rotating it. Eigenvalues determine stability in dynamic systems, natural frequencies in vibration analysis, and principal directions in stress and inertia problems.