List of basic properties of matrix operations:
- Matrix addition:
- Commutativity: (A + B = B + A)
- Associability: ( (A + B) + C = A + (B + C) )
- Presence of zero matrix: (A + 0 = A)
- 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)
- Scalar multiplication:
- ( c(A + B) = cA + cB )
- ( (c + d)A = cA + dA )
- (c(dA) = (cd)A)
- ( 1A = A )
- Transpose:
- ( (A^T)^T = A )
- ( (A + B)^T = A^T + B^T )
- ( (AB)^T = B^T A^T )
- ( (cA)^T = cA^T )
- 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.
- 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} )