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  1. 1Determinant of a 2×22\times 2 matrix
  2. 3Determinant, upper triangular
  3. 4Determinant of a 3×33\times 3 matrix
  4. 5Eigenvalues of a diagonal matrix
  5. 6Eigenvalues of a 2×22\times 2 matrix
  6. 7Eigenvalues of a 2×22\times 2 matrix
  7. 8Eigenvalues of a 2×22\times 2 matrix
  8. 9Eigenvalues of a 2×22\times 2 matrix
  9. 11Eigenvalues of a triangular matrix
  10. 13Inverse of a 2×22\times 2 matrix
  11. 15Inverse of lower triangular AA
  12. 16Solving a linear system
  13. 18Inconsistent linear system
  14. 19Cramer's rule, 2×22\times 2 system
  15. 23Rank by row reduction
  16. 24Rank of a 3×33\times 3 matrix
  17. 25Null space dimension
  18. 26Rank-nullity theorem
  19. 28Dot product of vectors
  20. 29Angle between two vectors
  21. 30Length of a vector
  22. 31Cross product of i\mathbf{i} and j\mathbf{j}
  23. 32Cross product of 3D vectors
  24. 33Projection of bb onto aa
  25. 34Projection of bb onto aa
  26. 35Checking linear independence
  27. 36Area via determinant
  28. 38Multiplying 2×22\times 2 matrices
  29. 39Squaring a 2×22\times 2 matrix
  30. 40Powers of a diagonal matrix
  31. 41Trace of a matrix
  32. 42Trace of A2A^2
  33. 44det⁡(2A)\det(2A) scaling rule
  34. 45Determinant of inverse
  35. 46Determinant of a product
  36. 52Determinant by row operations

40Powers of a diagonal matrix

Compute A10\displaystyle A^{10} for A=[2003]\displaystyle A = \begin{bmatrix}2 & 0\\0 & 3\end{bmatrix}.

For a diagonal matrix, raising it to a power just raises each diagonal entry to that power—no entries mix. So A^10 = diag(2^10, 3^10) = diag(1024, 59049). Check: A² = diag(4,9) matches 2² and 3².

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