Use the calculator
Use the calculator by choosing a square matrix size, filling every cell, and selecting Calculate. The tool shows the determinant and the row-elimination steps.
- Leave the default size of
3, or set Size (n × n) from1to8. - Fill every cell in Matrix A with a valid number or fraction.
- Select Calculate to display the result.
Enter valid matrix values
Use plain integers, signed numbers, decimals, or simple fractions such as 3/4 and -7/2.
- Do not leave a cell empty; empty cells are rejected.
- Do not enter expressions, variables, or other symbols.
- Use a nonzero denominator in every fraction.
The interface always creates an n × n square matrix, so it does not provide a way to enter a non-square matrix.
Read the result
The result shows Determinant, Matrix size, and Status. A non-integer determinant includes the exact fraction and a decimal approximation. The Elimination steps area lists the row operations and resulting matrix states.
A Singular result states det A = 0 and notes that A has no inverse. A non-singular result is labeled Non-singular.
- Copy result copies the entered matrix, determinant, matrix size, and status. It does not include the visible elimination steps.
- Download result (.txt) downloads those same fields as
matrix-determinant-result.txt.
Copy result depends on browser clipboard access. If access is unavailable, the page may show no error message.
How the determinant is calculated
The calculator uses row elimination to reduce the matrix to upper-triangular form:
- It finds a nonzero pivot in the current column.
- If needed, it swaps rows. Each swap changes the determinant sign.
- It eliminates entries below the pivot. These row combinations do not change the determinant.
- It multiplies the diagonal entries and applies the sign from the row swaps.
If no pivot is available in a required column, the matrix is singular and the determinant is exactly 0. Values are calculated as reduced exact fractions backed by BigInt.
Reset or change the matrix size
Changing Size (n × n) hides the result, clears errors, rebuilds the grid, and removes all existing entries. The control accepts sizes from 1 to 8; an entered value above 8 is clamped to 8.
Select Reset to rebuild the current-size grid and clear the result and error state.
Limits and local processing
- Only square matrices from
1 × 1through8 × 8can be entered through this interface. - The calculation script processes matrix values locally in the browser and does not upload them as part of the calculation.
- The page also contains analytics and a subscription form, so this does not mean that nothing on the page is sent to a server.
- The tool does not evaluate general numerical conditioning or provide broader stability analysis.
- Cell input is parsed as a number or simple fraction; it is not executed as code.
Frequently Asked Questions
What is the formula for a matrix determinant?
This calculator uses row elimination rather than cofactor expansion. It reduces A to upper-triangular form with row swaps and row combinations, then calculates det A from the signed product of the diagonal entries.
What size matrices can I use?
Any square matrix from 1 × 1 up to 8 × 8. The Size (n × n) control creates only square matrices.
How are results kept exact?
Every entry is stored as an exact fraction with an integer numerator and denominator reduced to lowest terms. A determinant such as 7/12 is shown exactly, with a decimal approximation alongside it.
How does it detect a singular matrix?
If a column has no nonzero entry available for a pivot at or below the current row, the matrix is reported as singular and its determinant is exactly 0. The calculation uses exact fractions, but it does not evaluate general numerical conditioning.
Does the calculator execute my input as code?
No. Each cell accepts only a plain number or a simple fraction such as 3/4. Other expressions and symbols are rejected rather than executed.