Theorem 1: Matrix addition is Commutative, i.e.
A+B = B+A for all comparable matrices A and B
Proof: Let A = [ aij]m x n B = [ bij]m x n
A+B = [ aij]m x n + [ bij]m x n
= [ aij + bij ]m x n (by definition of addition matrices)
= [ bij+aij ]mxn (since addition of numbers is commutative)
= [bij]mxn + [ aij]mxn = B+A
A+B = B+A
Theorem 2: Matrix addition is associative. i.e
(A+B)+C = A+(B+C) for all comparable matrices A,B,C.
Proof: Let A = [ aij]m x n , B = [ bij]m x n C = [ cij]m x n
(A+B)+C = ([ aij]m x n + [ bij]m x n) + [ cij]m x n
= [ aij + bij ]m x n + [ cij]m x n
=[( aij + bij)+ cij]m x n
=[aij + (bij + cij)]m x n
=[ aij]m x n + [(bij + cij)]m x n
Theorem 3: If A is an mxn matrix and O is an mxn null matrix, then
A+O =O+A =A
Proof: Let A = [ aij]m x n O = [ bij]m x n’
Where, bij = 0
A+O = [ aij]m x n + [ bij]m x n’
= [ aij + bij ]m x n
= [ aij + 0 ]m x n
= [ aij]m x n
A+O = A
O+A = A
For a given matrix A, its negative is obtained by replacing each element with its corresponding additive inverse.
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