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Orthogonal Matrix Checker

Enter the four entries of your 2x2 matrix to check if it is orthogonal.

How it is calculated

The matrix is read as rows [[a, b], [c, d]], so column 1 is (a, c) and column 2 is (b, d). Three numbers are computed: a² + c² (squared length of column 1), b² + d² (column 2) and a·b + c·d (dot product of the columns). The matrix is called orthogonal only if the first two are within 0.000001 of 1 and the dot product is within 0.000001 of 0. The determinant a·d − b·c is shown to four decimals: +1 means a rotation, −1 a reflection. The fixed tolerance is tight, so numbers rounded to three or four decimals usually fail. Only 2×2 matrices are handled, and the result does not depend on any country or year.

Worked example

Matrix with a = 0.6, b = −0.8, c = 0.8, d = 0.6. Column 1: 0.6² + 0.8² = 0.36 + 0.64 = 1.000000. Column 2: (−0.8)² + 0.6² = 0.64 + 0.36 = 1.000000. Dot product: 0.6×(−0.8) + 0.8×0.6 = −0.48 + 0.48 = 0.000000. All three tests pass, so the result is Orthogonal. Determinant = 0.6×0.6 − (−0.8)×0.8 = 0.36 + 0.64 = 1.0000, a pure rotation.

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Common mistakes

  • Entering a rounded rotation matrix such as cos 30° as 0.866 and sin 30° as 0.5. The squared column length is 0.999956, outside the 0.000001 tolerance, so the tool says Not orthogonal; type more decimals.
  • Entering the entries in column order instead of row order. For [[a, b], [c, d]], b is the top right entry and c is the bottom left; swapping them changes the transposed matrix.
  • Believing perpendicular columns are enough. The matrix [[1, 1], [−1, 1]] has dot product 0 but column length squared 2, so its columns are perpendicular but not unit length and the tool says Not orthogonal.
  • Expecting the determinant to decide it. A determinant of 1 can occur in a non-orthogonal matrix such as [[1, 1], [0, 1]].
  • Using the tool for 3×3 or larger matrices; the form has only four entries.

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.

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Comments (1)

  • Editorial team

    When the answer is Not orthogonal, read the checks line to see which test failed. If only the two column lengths fail while the dot product passes, the columns are perpendicular and dividing each column by its own length gives an orthogonal matrix. If the dot product fails, the columns are not perpendicular at all. Where do your matrices come from: graphics, robotics or a course? Tell us if you need 3 by 3.

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