Compute the determinant, transpose or inverse of a matrix up to 10×10 — with step-by-step results.
A matrix is a rectangular array of numbers, and three of the most common operations on it are the determinant, the transpose and the inverse. The determinant is a single number computed from a square matrix that tells you whether the matrix is invertible — if it is zero, the matrix is singular and has no inverse. The transpose swaps rows and columns: row i becomes column i and vice versa. The inverse, when it exists, is the matrix that multiplied by the original gives the identity.
This tool uses cofactor expansion for the determinant (recursive, expanding along the first row) and Gauss-Jordan elimination for the inverse (augmenting the matrix with the identity and row-reducing until the left half is the identity). Both algorithms are exact for integer inputs and produce floating-point results for decimals. The grid accepts matrices up to 10×10, which covers most educational and practical needs. Larger matrices require dedicated linear algebra software.
The result is shown as a formatted matrix that you can copy. The inverse display includes the determinant as a sanity check — if it is close to zero, the matrix is nearly singular and the inverse may be numerically unstable. Everything runs in your browser with no server computation.
Determinant (cofactor expansion): det(A) = Σ (-1)^(i+j) × a_ij × det(M_ij) where M_ij is the minor matrix. Inverse (Gauss-Jordan): augment [A | I], row-reduce to [I | A⁻¹].
10×10. The cofactor expansion algorithm is O(n!), so it becomes slow above 10. For large matrices, use a numerical computing tool.
Not in this version — it handles one matrix at a time. For multiplication, you would need a second input grid.
If the determinant is close to zero, the matrix is nearly singular and the inverse amplifies rounding errors. The result is mathematically correct but numerically fragile.
The determinant and inverse are defined only for square matrices. The transpose works on any shape — a 3×2 matrix becomes 2×3.
No. Everything is computed in JavaScript on your device.
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