Determinant and Trace Calculator

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 picks the method that shows the clearest steps for each size:

• \(2\times 2\): the direct rule \(\det(A) = a_{11}a_{22} - a_{12}a_{21}\) (i.e. \(ad - bc\)).
• \(3\times 3\) and \(4\times 4\): cofactor (Laplace) expansion along a row or column of your choice — pick it from the selector that appears above the results.
• \(n \ge 5\): elementary row operations⁴ reduce \(A\) to an upper-triangular matrix \(U\); then \(\det(A)\) is the product of the diagonal entries of \(U\) times the product of the elementary matrix determinants.