Determinant and Trace Calculator

Compute the determinant1 and trace2 of a square matrix.
Two fundamental scalar quantities that characterize square matrices.

Determinant:

  • Denoted \(\det(A)\) or \(|A|\)
  • Measures the volume scaling factor of the linear transformation3
  • \(\det(A) = 0\) if and only if \(A\) is singular (non-invertible)
  • For eigenvalues \(\lambda_1, \ldots, \lambda_n\): \(\det(A) = \lambda_1 \cdots \lambda_n\)

Trace:

  • Denoted \(\operatorname{tr}(A)\)
  • Sum of diagonal entries: \(\operatorname{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn}\)
  • For eigenvalues \(\lambda_1, \ldots, \lambda_n\): \(\operatorname{tr}(A) = \lambda_1 + \cdots + \lambda_n\)

This calculator uses elementary row operations4 to reduce \(A\) to an upper-triangular matrix \(U\), then computes \(\det(A)\) as the product of the diagonal entries of \(U\) times the product of the elementary matrix determinants.

Set matrix dimension

Enter the matrix size n and click Generate Matrix to create an n×n input grid.

Display values as:
1 — Set Up
Loads a pre-filled 3×3 matrix — ready to compute.
2 — Compute
Computes the determinant and trace of the matrix using elementary row operations.
Resets all dimensions, matrix entries, and results.