Determinant Calculator – det(A) for 2-2 to 8-8 (Fractions Supported)

Use this free Determinant Calculator to compute det(A) fast and accurately. Enter numbers or fractions like 3/4, pick matrix size up to 8�8, and optionally view step-by-step row reduction with partial pivoting for numerical stability. For 2�2 & 3�3, the tool also shows the closed-form formulas (including Sarrus for 3�3).

Determinant Calculator

Random integers in [-5, 9]

Tip: You can enter fractions like -7/3 or decimals. Commas are auto-converted.
det(A): �
Method: �
Paste matrix (CSV / whitespace). Example (3�3): 1,2,3\n4,5,6\n7,8,9

Note: The calculator uses Gaussian elimination with partial pivoting. If a pivot is zero (or near zero), it swaps rows to maintain stability. Determinant is the product of the upper-triangular diagonal times the row-swap sign.

What Is the Determinant?

The determinant det(A) is a scalar attached to a square matrix A that encodes geometric and algebraic information�most famously: a matrix is invertible iff det(A) ? 0. Geometrically, |det(A)| is the area/volume scale factor of the linear transformation defined by A, and its sign indicates orientation.

How This Calculator Computes det(A)

For sizes ?4, cofactor expansion grows explosively. This tool uses row-reduction (Gaussian elimination) with partial pivoting to reach an upper-triangular matrix U. Then det(A)=sign�?Uii, where sign flips for each row swap.

Formulas for 2�2 and 3�3 (Sarrus)

  • 2�2: det [[a,b],[c,d]] = ad ? bc
  • 3�3 (Sarrus): Extend the first two columns and sum forward diagonals minus backward diagonals.

Key Properties of Determinants

  • Row swap multiplies determinant by ?1; row scaling by k scales det by k.
  • Adding a multiple of one row to another leaves det unchanged (handy for elimination).
  • det(AB)=det(A)det(B); det(A?)=det(A).

Applications

  • Invertibility test & matrix inverse (via adjugate/Cramer�s Rule for small n).
  • Geometry: area/volume, orientation, change-of-variables in integrals (Jacobian).
  • Eigenstuff: characteristic polynomial uses determinants.

FAQ

Does it handle fractions exactly?

Fractions like 3/7 are parsed and evaluated to decimals before elimination. For exact symbolic algebra, use a CAS; this tool prioritizes speed and stability.

What sizes are supported?

2�2 up to 8�8.

Why partial pivoting?

It avoids dividing by tiny pivots, which reduces numerical error.