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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

43Characteristic polynomial, 2×22\times 2

Find the characteristic polynomial of A=[1234]\displaystyle A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}.

The characteristic polynomial is det(A−λI). Subtracting λ from the diagonal and expanding gives λ² − 5λ − 2. Check: trace(A) = 1+4 = 5 matches the −5λ coefficient, and det(A) = 1·4 − 2·3 = −2 matches the constant term.

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