Matrix Calculator: Determinant, Inverse, Multiply & RREF

A free matrix calculator for matrices up to 6 × 6. Paste or type your matrices, pick an operation and get the determinant, inverse, product, transpose, rank, trace, powers or reduced row echelon form, with answers shown as exact fractions where possible.

Matrix AOne row per line; separate entries with spaces or commas. Fractions like 1/2 work.

Matrix B (for two-matrix operations)

Operation

Power or scalar (k)

Matrix formulas

Multiplication (AB)ᵢⱼ = Σ Aᵢₖ·Bₖⱼ needs columns of A = rows of B 2×2 determinant det [a b; c d] = ad − bc 2×2 inverse A⁻¹ = (1 / (ad − bc)) · [d −b; −c a] 3×3 determinant a(ei − fh) − b(di − fg) + c(dh − eg) Inverse exists only when det(A) ≠ 0 Rank number of non-zero rows in RREF 2×2 eigenvalues λ = (tr ± √(tr² − 4·det)) / 2

How the calculator works

Determinants, inverses, rank and RREF are computed with Gauss–Jordan elimination using partial pivoting for numerical stability, then converted back to fractions. For the inverse, the calculator row-reduces the augmented matrix [A | I] until the left side becomes I; the right side is then A⁻¹. A matrix whose determinant is 0 is singular and has no inverse.

Worked example

For A = [2 1 3; 0 −1 4; 1 2 0], expanding along the first row gives det(A) = 2(0 − 8) − 1(0 − 4) + 3(0 + 1) = −16 + 4 + 3 = −9. Because the determinant is not zero, A is invertible and its rank is 3.

Remember that matrix multiplication is not commutative: A × B is usually different from B × A. Use the swap button to compare.

Frequently asked questions

How do I find the determinant of a matrix?

Type the square matrix into A, one row per line, choose Determinant of A and press Calculate. The calculator works for any size up to 6 × 6 and shows the exact value.

How do I find the inverse of a matrix?

Enter a square matrix in A and choose Inverse of A. If the determinant is zero the matrix is singular and has no inverse; otherwise you get A⁻¹ in exact fractions plus a check that A × A⁻¹ = I.

Why can’t I multiply my two matrices?

For A × B the number of columns in A must equal the number of rows in B. A 2 × 3 matrix can multiply a 3 × 4 matrix (giving 2 × 4), but not a 2 × 3 one.

What is RREF?

Reduced row echelon form is the simplest matrix you can reach with row operations: each leading entry is 1 and is the only non-zero number in its column. It shows the rank and solves linear systems.

Can I enter decimals and fractions?

Yes. Entries like 0.5, −3 and 2/3 all work, and results are converted back to simple fractions when they are exact.

Related tools

Free educational tool by HawkInc. Results are calculated in your browser and rounded for display; check critical work by hand or with a second method.