Linear Algebra Calculators

How a Matrix's Determinant Is Calculated

A matrix's determinant is computed by reducing it to upper-triangular form and multiplying the diagonal — far more efficient than the textbook cofactor-expansion method, and the single number that decides whether the matrix even has an inverse.

How a Matrix's Inverse Is Calculated

A matrix's inverse is found by row-reducing it alongside the identity matrix until the original matrix becomes the identity — at which point the identity's starting position has transformed into the inverse, sitting right where the answer needs to be.

How the Cross Product of Two 3D Vectors Is Calculated

The cross product of two 3D vectors is itself a vector — perpendicular to both inputs, with a length equal to the area of the parallelogram they span — computed from a specific pattern of cross-multiplied components.

How the Dot Product and Angle Between Two Vectors Are Calculated

The dot product multiplies each pair of matching components and adds the results, and rearranging that same formula gives the angle between the two vectors — a single calculation that tells you both how aligned two vectors are and by how much.

Why Multiplying a Matrix by Its Real Inverse Gives Back the Identity Matrix

Computing a matrix's real inverse and then multiplying the two together, using two entirely separate calculators built on different algorithms, returns exactly the identity matrix — the definition of "inverse," confirmed with real numbers rather than taken on faith.