Matrix Calculator
Free online matrix calculator: determinant, inverse, rank, trace, transpose, RREF and eigenvalues for matrices up to 6×6, plus A±B, A×B and B×A. Runs entirely in your browser.
What does the determinant of a matrix mean?
The determinant is a single number computed from a square matrix. It measures how the matrix scales area or volume, and its sign shows whether orientation is preserved. A determinant of 0 means the matrix is singular and has no inverse.
When does a matrix have no inverse?
A square matrix is invertible only when its determinant is not zero. Such matrices are called singular: their rows or columns are linearly dependent. This tool shows a friendly message instead of an error when you try to invert a singular matrix.
What do rank and trace tell me?
Rank is the number of linearly independent rows or columns — it shows how much real information the matrix carries. Trace is the sum of the diagonal entries and appears in eigenvalue theory, geometry and statistics.
What is the reduced row echelon form (RREF) used for?
RREF is the simplest form reachable with row operations. It makes solving linear systems trivial, reveals the pivot columns and shows the rank at a glance. It works on any shape from 1×1 to 6×6, not only square matrices.
How are the eigenvalues calculated?
Eigenvalues are the numbers λ for which A·x = λ·x has a nonzero solution. This tool computes them numerically: real eigenvalues appear as plain numbers and complex pairs in the a + bi form. Eigenvalues are only defined for square matrices.
Can I enter fractions like 1/2?
Yes. Every cell accepts integers (3), decimals (1.5) and fractions (1/2, -3/4). Cells that are empty or contain invalid text are highlighted in red so you can fix them quickly.
Which operations work on non-square matrices?
Transpose, RREF and rank work for any shape from 1×1 to 6×6. Determinant, trace, inverse and eigenvalues are defined only for square matrices, and A×B requires the column count of A to match the row count of B.
Is my matrix data uploaded anywhere?
No. All calculations run locally in your browser with a bundled math library. Nothing you type leaves your device and no account is needed.
What is this matrix calculator useful for?
It is handy for checking linear algebra homework, verifying hand-computed determinants and inverses, exploring rank and RREF, solving systems of linear equations, or quickly testing properties of small matrices.
How to Use Matrix Calculator
This matrix calculator analyzes a single square or rectangular matrix (determinant, rank, trace, inverse, transpose, reduced row echelon form and eigenvalues) and also performs operations on a pair of matrices A and B: A+B, A−B, A×B, B×A, det(A), det(B) and the inverses. Everything is computed locally in your browser — nothing is uploaded.
- Pick a mode with the tabs: "Single matrix" analyzes one matrix, "A · B operations" gives you two editable matrices and their combinations.
- Set the size freely from 1×1 up to 6×6 with the Rows and Columns steppers — the grid keeps the values you already typed when it grows or shrinks.
- Type numbers into the cells: integers (3), decimals (1.5, 2.25) and fractions (1/2, -3/4) all work. Invalid or empty cells are highlighted in red.
- Results update as you type: scalars (determinant, rank, trace, eigenvalues) appear as bold values and matrices as formatted tables. Square-only operations explain themselves when the matrix is rectangular, and a singular matrix shows a clear note instead of an error.
- Use the 2×2 or 3×3 example buttons to load ready-made matrices, Clear to empty the grid, and the small copy icon on any card to copy the result as LaTeX-style text (\begin{bmatrix} … \end{bmatrix}).
For A×B the column count of A must equal the row count of B, while B×A needs the reverse — different shapes give you both products. Determinant of 0 means no inverse exists; lower the rank below the matrix size is the quickest way to construct such a case on purpose.
Related Tools
- Math Expression Calculator Evaluate complex math expressions online with variables, 20+ functions, and real-time calculations. Perfect for algebra, trigonometry, and scientific computing.
- Scientific Calculator Free online scientific calculator with trig, log, and exponential functions. Switch between degree and radian modes instantly.
- Calorie Calculator Free calorie calculator. Find your BMR and daily calorie needs (TDEE), plus target calories for weight loss or gain with a protein, carb and fat breakdown - works offline in your browser.
- Arithmetic Sequence Calculator Calculate nth term, sum, and generate complete arithmetic sequences instantly. Free online tool with 100% privacy - all calculations happen locally in your browser.
- GCD/LCM Calculator Calculate the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of two or more numbers.
- Prime Number Checker Check if a number is prime instantly. Find prime factors, nearest primes, and the nth prime number online.
- Factorial Calculator Calculate n! instantly for any non-negative integer. Free online factorial calculator with BigInt support for arbitrarily large results.
- Fibonacci Generator Generate Fibonacci numbers and sequences instantly. Calculate the nth Fibonacci number or generate a full sequence with BigInt support for arbitrarily large numbers.