Matrix Calculator
Multiplies, inverts, transposes and row-reduces matrices in exact rational arithmetic, and finds determinant, rank and trace.
Multiply, invert, transpose and row-reduce matrices, and read off the determinant, rank and trace. Every entry is an exact rational rather than a floating point number, so a singular matrix has a determinant of exactly 0 and the inverse of an integer matrix comes back as 1/3 rather than 0.3333333333333333.
Matrix A
One row per line. Entries separated by spaces, commas or tabs. Fractions like 1/3 and decimals both work.
Matrix B
Operation
Result
2 1 4 3
About matrix A
shape
2 × 2
rank
2
determinant
-2
trace
5
Eigenvalues are deliberately not offered. Finding them means solving the characteristic polynomial, which past 4 × 4 has no closed form and needs an iterative numerical method whose answers are usually irrational — nothing the exact representation used here can express. A tool that printed floating-point eigenvalues would be making a different promise than this one does.
How to use it
Type the matrix
One row per line, entries separated by spaces, commas or tabs. Fractions like 1/3 are accepted as they are and stay exact.
Pick the operation
Some need a second matrix and some do not. Multiplication needs the columns of A to match the rows of B, and says so plainly when they do not.
Read the properties panel
Shape, rank, determinant and trace for A. Rank below the number of rows is why the inverse does not exist, and reading it saves guessing.
Common questions
Why are eigenvalues not offered?
Finding them means solving the characteristic polynomial, which past 4 × 4 has no closed form and needs an iterative numerical method whose answers are usually irrational — nothing the exact rational representation used here can express. A tool that printed floating-point eigenvalues would be making a different promise than this one does.
The determinant is 0 but my other calculator says 1e-16
That calculator is using floating point, where the rounding of each step leaves a residue. Here the arithmetic is exact, so a singular matrix gives exactly zero and you do not have to decide whether 3.6 × 10⁻¹⁶ counts as zero.
What is RREF for?
Reduced row echelon form solves a linear system, finds the rank and exposes which variables are free. It is what elimination is doing under any of the other operations, made visible.
Feed it from
Equation Solver
Solves linear, quadratic and cubic equations with the working shown, finds real roots of other equations numerically, and solves linear systems by elimination.
Big Number Calculator
Performs exact arithmetic on numbers far beyond 64 bits, including power, modulo and factorial.
Unit Converter
Converts length, weight, area, volume, temperature, speed, pressure and data units, including Korean units.
Number Base Converter
Converts between binary, octal, decimal, hexadecimal and any base up to 36, with bitwise views.
Next
These accept what this tool produces, so the result can go straight in.
Big Number Calculator
Performs exact arithmetic on numbers far beyond 64 bits, including power, modulo and factorial.
Equation Solver
Solves linear, quadratic and cubic equations with the working shown, finds real roots of other equations numerically, and solves linear systems by elimination.
Descriptive Statistics
Computes mean, median, mode, variance, standard deviation, quartiles and outliers from pasted numbers.
CSV Cleaner
Trims cells, normalises headers, fixes number and date formats and reports rows with the wrong column count.
CSV Join
Joins two tables on a key column with inner, left, right or outer semantics and shows unmatched rows.
CSV Viewer
Opens a large delimited file with type detection, column stats and sorting, without loading it all into memory.