Properties of Matrix Operations   Properties of Addition The basic properties of addition for real numbers also hold true for matrices.  Let A, B and C be m x n matrices A + B  =  B + A    commutative A + (B + C)  =  (A + B) + C    associative There is a unique m x n matrix O with         A + O  =  A        additive identity For any  m x n matrix A there is an m x n matrix B (called -A) with        A + B  =  O        additive inverse   The proofs are all similar.  We will prove the first property.   Proof of Property 1 We have         (A + B)ij  =  Aij + Bij     definition of addition of matrices         =  Bij + Aij         commutative property of addition for real numbers        =  (B + A)ij       definition of addition of matrices Notice that the zero matrix is different for different m and n.  For example Properties of Matrix Multiplication Unlike matrix addition, the properties of multiplication of real numbers do not all generalize to matrices.  Matrices rarely commute even if AB and BA are both defined.  There often is no multiplicative inverse of a matrix, even if the matrix is a square matrix.  There are a few properties of multiplication of real numbers that generalize to matrices.  We state them now. Let A, B and C be matrices of dimensions such that the following are defined.  Then A(BC)  =  (AB)C                 associative A(B + C)  =  AB + AC        distributive (A + B)C  =  AC + BC        distributive There are unique matrices Im and In with         Im A  =  A In  =  A        multiplicative identity We will often omit the subscript and write I for the identity matrix.  The identity matrix is a square scalar matrix with 1's along the diagonal.  For example We will prove the second property and leave the rest for you.   Proof of Property 2 Again we show that the general element of the left hand side is the same as the right hand side.  We have         (A(B + C))ij  = S(Aik(B + C)kj)        definition of matrix multiplication         =  S(Aik(Bkj + Ckj))        definition of matrix addition         =  S(AikBkj + AikCkj)       distributive property of the real numbers         =  S AikBkj + S AikCkj     commutative property of the real numbers         =  (AB)ij + (AC)ij        definition of matrix multiplication where the sum is taken from 1 to k. Example We will demonstrate property 1 with We have so that We have so that Properties of Scalar Multiplication Since we can multiply a matrix by a scalar, we can investigate the properties that this multiplication has.  All of the properties of multiplication of real numbers generalize.  In particular, we have Let r and s be real numbers and A and B be matrices.  Then r(sA)  =  (rs)A    (r + s)A  =  rA + sA r(A + B)  =  rA + rB A(rB)  =  r(AB)  =  (rA)B   We will prove property 3 and leave the rest for you.  We have         (r(A + B))ij  =  (r)(A + B)ij          definition of scalar multiplication         =  (r)(Aij + Bij)        definition of addition of matrices         =  rAij + rBij        distributive property of the real numbers         =  (rA)ij + (rB)ij        definition of scalar multiplication         =  (rA + rB)ij        definition of addition of matrices Properties of the Transpose of a Matrix Recall that the transpose of a matrix is the operation of switching rows and columns.  We state the following properties.  We proved the first property in the last section. Let r be a real number and A and B be matrices.  Then (AT)T  =  A (A + B)T  =  AT + BT (AB)T  =  BTAT (rA)T  =  rAT Back to the Linear Algebra Home Page