Matrix Calculator
Add, subtract, multiply and transpose 2×2 or 3×3 matrices, plus find the determinant and inverse with the full cofactor-expansion trace — minors, signed cofactors, the adjugate and every step shown as exact fractions.
Result
A matrix calculator earns its keep only if it shows the actual arithmetic, not just a boxed answer — so this one does. Pick a 2×2 or 3×3 size and an operation, and every intermediate number appears: addition and subtraction show each cᵢⱼ = aᵢⱼ ± bᵢⱼ substitution, multiplication shows the full row·column dot product for every output cell (c11 = a11×b11 + a12×b21 + a13×b31, and so on), and transpose shows exactly which entry moved where. Nothing is computed silently in the background.
The determinant and inverse are where a matrix calculator usually turns into a black box, so this is the tool's centerpiece. For a 3×3 matrix the determinant is expanded along the first row exactly the way a textbook does it — det(A) = a11×M11 − a12×M12 + a13×M13 — and each 2×2 minor Mᵢⱼ gets its own worked-out ad − bc line. The inverse goes one step further: all nine minors become signed cofactors following the +/−/+ checkerboard pattern (−1)^(i+j), the cofactor matrix is transposed into the adjugate, and every entry of the adjugate is divided by the determinant to produce A⁻¹ — with every one of those steps printed, not skipped.
All of this arithmetic is exact. Matrix entries are parsed as digits over a power of ten and combined with BigInt-backed fraction arithmetic (reduced with the Euclidean algorithm at every step), so a determinant of 6 stays 6, and an inverse entry that would round to −0.666667 in floating point instead prints as the exact fraction −2/3. When the determinant is zero, the tool doesn't divide by it — it shows the determinant calculation that produced zero and states plainly that the matrix is singular and has no inverse.
Everything runs locally in your browser: nothing is uploaded, there is no registration, and you can run as many matrices as you like on desktop or mobile. It's built for checking linear algebra homework step by step, verifying a hand calculation of a 3×3 inverse, or working through Cramer's-rule-style problems where you need the exact cofactor matrix rather than a rounded decimal.
Your files and inputs never leave this device. Processing runs completely offline. 100% Secure.
Accepted Formats & Rules
- Handles 2×2 and 3×3 matrices for addition, subtraction, multiplication, transpose, determinant and inverse.
- The 3×3 determinant and inverse expand the real cofactor method: nine minors, the +/−/+ sign checkerboard, the adjugate matrix, then division by the determinant.
- All arithmetic — sums, dot products, minors, cofactors, the final division — is done with exact reduced fractions, never floating-point decimals.
Key Benefits
Use Cases
- • Check a 3×3 inverse computed by hand for a linear algebra course: enter the matrix, switch to Inverse, and compare every minor, cofactor and adjugate entry against your own scratch work instead of only checking the final answer.
- • Solve a small system of linear equations by hand via the inverse matrix method (x = A⁻¹b): find A⁻¹ here as exact fractions, then multiply it by the constants column yourself with numbers that are guaranteed not to be rounded.
- • Verify that two 2×2 transformation matrices (e.g. rotation and scaling matrices in a graphics or robotics assignment) multiply into the expected composite matrix, with the row·column dot product for every cell shown.
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