List of mathematical formulas: “Properties of matrix operations.

List of basic properties of matrix operations:

  1. Matrix addition:
  • Commutativity: (A + B = B + A)
  • Associability: ( (A + B) + C = A + (B + C) )
  • Presence of zero matrix: (A + 0 = A)
  1. Matrix multiplication:
  • Non-commutativity: ( AB \neq BA ) (in general)
  • Associability: ( A(BC) = (AB)C )
  • Distributivity with respect to addition: (A(B + C) = AB + AC) and ((A + B)C = AC + BC)
  1. Scalar multiplication:
  • ( c(A + B) = cA + cB )
  • ( (c + d)A = cA + dA )
  • (c(dA) = (cd)A)
  • ( 1A = A )
  1. Transpose:
  • ( (A^T)^T = A )
  • ( (A + B)^T = A^T + B^T )
  • ( (AB)^T = B^T A^T )
  • ( (cA)^T = cA^T )
  1. Determinant:
  • ( \det(AB) = \det(A) \cdot \det(B) )
  • ( \det(A^T) = \det(A) )
  • If ( A ) is an upper or lower triangular matrix, then ( \det(A) ) is equal to the product of the diagonal elements.
  1. Inverse matrix:
  • ( AA^{-1} = I ) and ( A^{-1} A = I ) (where ( I ) is the identity matrix)
  • ( (A^T)^{-1} = (A^{-1})^T )
  • ( (AB)^{-1} = B^{-1} A^{-1} )

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