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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

10Eigenvalues of a 3×33\times 3 matrix

Find the eigenvalues of A=[120210003]\displaystyle A = \begin{bmatrix}1 & 2 & 0\\2 & 1 & 0\\0 & 0 & 3\end{bmatrix}.

Solve det(A−λI)=0. Expanding along the bottom row (two zeros make it easy) gives (3−λ)[(1−λ)²−4]=0, which factors to (3−λ)(3−λ)(−1−λ)=0. So λ = 3, 3, −1. Check: sum = 5 = trace of A.

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