**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

Then,

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

Hence,

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

Then,

(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

= A+(B+C)

Hence,

(A+B)+C= A+(B+C)

**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

Then,

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

Therefore,

A+O = A

Similarly,

O+A = A

Hence,

A+O=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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