Orthogonal Matrix Checker

About this calculator

A square matrix Q is orthogonal when its transpose is also its inverse: Q^T Q = I, the identity matrix. Equivalently, the columns (and rows) of Q form an orthonormal set — each column has unit length and every pair of distinct columns is perpendicular, with dot product zero.

Checking this condition for a 2×2 matrix means testing that both columns have length 1 and their dot product is 0; the calculator also reports the determinant, which for any orthogonal matrix must equal +1 (a pure rotation) or −1 (a rotation combined with a reflection).

Orthogonal matrices preserve lengths and angles, which is why they represent rotations and reflections in computer graphics, robotics, and physics, and why they appear as the Q factor in QR decomposition used to solve least-squares problems numerically.