Matrix Calculator
Calculate matrix determinant, inverse, RREF, rank, multiplication and powers with exact fractions and row-operation steps. Free online matrix calculator.
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Part of the toolkit
How to use Matrix Calculator
- 1Set the size of matrix A (and B if you need two matrices) up to 6×6, and type the entries — integers, decimals or fractions like 1/3
- 2Press an operation: A + B, A − B, A × B, determinant, inverse, transpose, RREF, rank, trace, power or k × A
- 3Read the exact answer as fractions (or switch to decimals), open the row operations to see each step, and copy the result
Frequently asked questions
How do you find the determinant of a 3×3 matrix?▾
Expand along the first row: det = a(ei − fh) − b(di − fg) + c(dh − eg) for rows (a b c), (d e f), (g h i). For larger matrices it's faster to row-reduce to triangular form and multiply the pivots, changing the sign for each row swap — which is what this calculator does.
How do you find the inverse of a matrix?▾
Write [A | I] and row-reduce until the left half becomes the identity; the right half is then A⁻¹. For a 2×2 matrix (a b; c d) the inverse is 1/(ad − bc) × (d −b; −c a). A matrix has an inverse only if its determinant is not zero.
What is RREF (reduced row echelon form)?▾
It's the simplest form a matrix can be row-reduced to: every leading entry is 1, it's the only non-zero number in its column, and each leading 1 is to the right of the one above. Row-reducing an augmented matrix to RREF solves a system of linear equations.
When can two matrices be multiplied?▾
A × B works when the number of columns of A equals the number of rows of B. A 2×3 matrix times a 3×4 matrix gives a 2×4 result. Matrix multiplication is not commutative: A × B is usually different from B × A.
What does a determinant of zero mean?▾
The matrix is singular: its rows (and columns) are linearly dependent, it has no inverse, and a system Ax = b has either no solution or infinitely many. Geometrically it squashes space into a lower dimension.
What is the rank of a matrix?▾
The number of linearly independent rows (equal to the number of independent columns). It's the count of pivots after row reduction. A square n×n matrix is invertible exactly when its rank is n.
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