How this calculator works
This summary page takes one square matrix and reports the quantities students check most often. The determinant tests invertibility, the trace adds the diagonal, the rank counts independent rows, the transpose swaps rows and columns, and the characteristic polynomial has the eigenvalues as roots. The inverse is shown only when the determinant is non-zero. Each property also has a dedicated page with more detail and options.
Formula
For an n by n matrix A: det(A), tr(A) = sum of aii, rank(A), A^T, A inverse = adj(A)/det(A), and p(λ) = det(λI - A).
Worked example
The matrix with rows 2, 1 and 1, 3 has determinant 5, trace 5, rank 2, transpose 2, 1; 1, 3, inverse 0.6, -0.2; -0.2, 0.4, and characteristic polynomial λ^2 - 5λ + 5. The 3 by 3 identity has determinant 1, trace 3, rank 3 and polynomial (λ - 1)^3.
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Frequently asked questions
Can I use a rectangular matrix?
This page is for square matrices because the determinant, trace and inverse need a square shape. Use the rank, null space or transpose pages for rectangular input.
Why is the inverse missing for some matrices?
A singular matrix has determinant zero, so division by the determinant is impossible. The page reports that no inverse exists instead of a numerical approximation.
Where can I find the eigenvalues?
The characteristic polynomial is shown here. Open the eigenvalue page for its roots and the matching eigenvectors, or the diagonalization page for the full decomposition.