AnyTool Matrix Calculator computes the determinant by reducing the matrix to upper-triangular form with LU decomposition and partial pivoting, then multiplying the pivots (adjusting the sign for each row swap) — an O(n³) method that is far more stable than recursive cofactor expansion. The inverse is found by Gauss–Jordan elimination on the augmented matrix [A | I]: row operations turn the left block into the identity, leaving the inverse on the right. If a pivot column is numerically zero the matrix is singular and the tool reports that no inverse exists instead of dividing by zero.

Matrix addition and subtraction work entry by entry, so both matrices must have exactly the same number of rows and columns. Multiplication A × B is defined only when the number of columns of A equals the number of rows of B, producing a matrix with the rows of A and the columns of B. AnyTool checks these dimensions before computing and returns a clear message — for example explaining the exact sizes involved — rather than silently producing a wrong answer.

Only square matrices can have an inverse, and only when their determinant is non-zero. A determinant of zero means the rows (and columns) are linearly dependent — the matrix collapses some directions to zero, so the transformation cannot be undone. AnyTool treats a determinant whose magnitude is below a tiny tolerance as singular to avoid meaningless results from floating-point noise, and reports that no inverse exists. You can confirm this with the rank: a singular n×n matrix has rank less than n.

AnyTool Matrix Calculator performs all parsing and arithmetic locally in JavaScript on a pure, unit-tested engine, so the numbers you enter are never sent to a server, there is no account or tracking, and the tool keeps working without a connection once it has loaded. Results are rounded for display to remove binary noise but are computed at full double precision.