3x3 Determinant Calculator

This calculator finds the determinant of a 3 by 3 matrix from its nine entries, and tells you whether the matrix is invertible. The determinant is one of the most important numbers associated with a square matrix: it captures, in a single value, key information about the linear transformation the matrix represents. Geometrically, the determinant of a 3 by 3 matrix is the signed volume of the parallelepiped formed by its three column vectors, so a determinant of zero means those vectors are flattened into a plane or line and the matrix collapses space. Practically, the determinant tells you whether a matrix can be inverted and whether a system of three linear equations has a unique solution: a non-zero determinant means yes, while a zero determinant means the matrix is singular and the system has either no solution or infinitely many. This tool computes it cleanly. You enter the nine numbers of the matrix in three rows, and the calculator returns the determinant using cofactor expansion along the top row, states whether the matrix is invertible, and shows the trace, the sum of the diagonal entries. The results update as you type, so you can see how changing an entry affects the determinant and watch it pass through zero as the matrix becomes singular. Use it to check linear algebra homework, to test whether a system has a unique solution, or to find the determinant as a step toward an inverse or Cramer's rule. The 3 by 3 case is where determinants first become genuinely fiddly to do by hand, with six product terms to track, which is exactly where a calculator helps. The result is exact for your inputs.

For the matrix [a b c; d e f; g h i], det = a(ei - fh) - b(di - fg) + c(dh - eg). A non-zero determinant means the matrix is invertible.