Matrix Calculator.

Formula or algorithm and variables.

Formula or algorithm and variables

det(A) uses elimination; (AB)ᵢⱼ = Σ AᵢₖBₖⱼ. Matrix A and Matrix B each have independently selected rectangular dimensions. Multiplication requires A columns to equal B rows; addition and subtraction require equal dimensions. Matrix addition and subtraction combine matching cells; multiplication takes row-by-column dot products; determinants and inverses use elimination. Start by identifying the mathematical object represented by each input rather than treating every number as interchangeable. Keep the chosen mode visible when a tool has more than one interpretation. The calculation preserves exact integer arithmetic where the model is discrete and uses controlled numeric rounding where the result is real-valued. This makes the path from the input to the displayed answer inspectable and gives you a repeatable way to check a hand calculation.

Worked example

The determinant of [[1,2],[3,4]] is 1×4 − 2×3 = −2, so the matrix is nonsingular and invertible. Reproduce the example by writing the formula, substituting the values, and retaining units or place-value labels through each step. For a list-based tool, sort or group the values before checking the result; for a base-conversion tool, expand each digit by its place value. When the result is a large integer, compare the exact string rather than a rounded approximation. A worked example is a verification aid, not a default recommendation, and it should be replaced with the values from the problem you are actually solving.

How to interpret the result

The calculator returns a formatted matrix, determinant, or inverse and states when dimensions or singularity make the requested operation undefined. Read the primary result together with its supporting details and the selected mode. A mathematically correct result can still answer the wrong question if the inputs describe a different population, representation, unit, ordering rule, or sample space. Compare scenarios only when the definitions and assumptions remain the same. Use exact notation for factors, matrices, repeating decimals, and integer bases; use the displayed precision for real-number summaries and avoid implying more certainty than the inputs support.

Validation and edge cases

Matrices must be rectangular, contain finite numbers, have matching dimensions for addition/subtraction, and have compatible dimensions for multiplication. Blank fields, malformed tokens, non-finite values, incompatible dimensions, invalid divisors, impossible probability ranges, and out-of-domain operations are rejected explicitly. Boundary values such as zero, one, a prime, a singular matrix, an empty remainder cycle, or a one-term sequence are tested according to the selected operation. If the output seems surprising, first verify the mode, the input order, the representation width, and whether a quantity was intended to be a count or a measurement.

Limitations and responsible use

Floating-point elimination can accumulate rounding error. The tool is limited to matrices up to 6×6 and is not a symbolic algebra or numerical linear-algebra certification system. This page is an explanatory calculator, not a substitute for a statistical design, numerical-analysis package, software type specification, laboratory model, or professional review. Results are processed locally in the browser and no entered values are sent to a calculation service. Keep the original inputs, mode, and assumptions with any copied result. For decisions involving health, finance, safety, experiments, or production systems, verify the model with current authoritative guidance and an independent method before acting.

Check assumptions and sources.

Floating-point elimination can accumulate rounding error. The tool is limited to matrices up to 6×6 and is not a symbolic algebra or numerical linear-algebra certification system.

NIST Engineering Statistics Handbook