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  1. 23×33\times 3 determinant expansion
  2. 10Eigenvalues of a 3×33\times 3 matrix
  3. 12Eigenvector for eigenvalue 3
  4. 14Inverse of a 2×22\times 2 matrix
  5. 17Solving a 3×33\times 3 linear system
  6. 20System with no solution
  7. 21Singular matrix, det=0
  8. 22Eigenvalue zero of a matrix
  9. 27Null space of a matrix
  10. 37Volume via scalar triple product
  11. 43Characteristic polynomial, 2×22\times 2
  12. 47Orthogonal matrix check
  13. 48Gram-Schmidt second vector
  14. 49Least-squares line fit
  15. 50Coordinates in a new basis
  16. 51Distance point to plane

47Orthogonal matrix check

Is A=[0110]\displaystyle A = \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix} orthogonal? Find det⁡A\displaystyle \det A.

A is orthogonal since AᵗA = I (swapping rows into columns and multiplying back gives the identity). Its determinant is det A = (0)(0) − (1)(1) = −1, which fits the rule that orthogonal matrices always have determinant ±1.

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