silemus.
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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

12Eigenvector for eigenvalue 3

A=[2112]\displaystyle A = \begin{bmatrix}2 & 1\\1 & 2\end{bmatrix} has eigenvalue 3\displaystyle 3. Find an eigenvector for it, scaled so its first entry is 1\displaystyle 1.

Solve (A − 3I)x = 0: subtracting 3 from the diagonal gives [[-1,1],[1,-1]], and both rows say y = x. Scaling so the first entry is 1 gives the eigenvector (1, 1). Check: A(1,1)ᵗ = (3,3)ᵗ = 3(1,1)ᵗ ✓.

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