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  1. 1Determinant of a 2×22\times 2 matrix
  2. 23×33\times 3 determinant expansion
  3. 3Determinant, upper triangular
  4. 4Determinant of a 3×33\times 3 matrix
  5. 5Eigenvalues of a diagonal matrix
  6. 6Eigenvalues of a 2×22\times 2 matrix
  7. 7Eigenvalues of a 2×22\times 2 matrix
  8. 8Eigenvalues of a 2×22\times 2 matrix
  9. 9Eigenvalues of a 2×22\times 2 matrix
  10. 10Eigenvalues of a 3×33\times 3 matrix
  11. 11Eigenvalues of a triangular matrix
  12. 12Eigenvector for eigenvalue 3
  13. 13Inverse of a 2×22\times 2 matrix
  14. 14Inverse of a 2×22\times 2 matrix
  15. 15Inverse of lower triangular AA
  16. 16Solving a linear system
  17. 17Solving a 3×33\times 3 linear system
  18. 18Inconsistent linear system
  19. 19Cramer's rule, 2×22\times 2 system
  20. 20System with no solution
  21. 21Singular matrix, det=0
  22. 22Eigenvalue zero of a matrix
  23. 23Rank by row reduction
  24. 24Rank of a 3×33\times 3 matrix
  25. 25Null space dimension
  26. 26Rank-nullity theorem
  27. 27Null space of a matrix
  28. 28Dot product of vectors
  29. 29Angle between two vectors
  30. 30Length of a vector
  31. 31Cross product of i\mathbf{i} and j\mathbf{j}
  32. 32Cross product of 3D vectors
  33. 33Projection of bb onto aa
  34. 34Projection of bb onto aa
  35. 35Checking linear independence
  36. 36Area via determinant
  37. 37Volume via scalar triple product
  38. 38Multiplying 2×22\times 2 matrices
  39. 39Squaring a 2×22\times 2 matrix
  40. 40Powers of a diagonal matrix
  41. 41Trace of a matrix
  42. 42Trace of A2A^2
  43. 43Characteristic polynomial, 2×22\times 2
  44. 44det⁡(2A)\det(2A) scaling rule
  45. 45Determinant of inverse
  46. 46Determinant of a product
  47. 47Orthogonal matrix check
  48. 48Gram-Schmidt second vector
  49. 49Least-squares line fit
  50. 50Coordinates in a new basis
  51. 51Distance point to plane
  52. 52Determinant by row operations

24Rank of a 3×33\times 3 matrix

Find the rank of A=[102013115]\displaystyle A = \begin{bmatrix}1 & 0 & 2\\0 & 1 & 3\\1 & 1 & 5\end{bmatrix}.

Row reduce A: R3 − R1 then R3 − R2 turns row 3 into all zeros, leaving two pivot rows. So rank(A) = 2. Check: row 3 of the original, (1,1,5), equals row1+row2, confirming it was dependent.

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