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

22Eigenvalue zero of a matrix

For which values of k\displaystyle k does A=[k11k]\displaystyle A = \begin{bmatrix}k & 1\\1 & k\end{bmatrix} have 0\displaystyle 0 as an eigenvalue?

Zero is an eigenvalue exactly when A is singular, so det(A) = 0. Here det(A) = k² − 1 = (k−1)(k+1), giving k = 1 or k = −1. Check k=1: the matrix is all 1's, with determinant 1·1 − 1·1 = 0.

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