Properties of Matrix Addition
If A = [ aij ] and B = [ bij ] are two matrices of the same order, say m × n. Then, the difference of the two matrices A and B is defined as a matrix C = [ cij ] m × n, where cij = aij- bij, for all possible values of i and j.
The two matrices have to be of the same order. if A and B are not of the same order, then A- B is not defined
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Solution:
Properties of Matrix Addition
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Commutative Law Associative Law Existence of additive identity The existence of additive inverse
Commutative Law: If A = [ aij ], B = [ bij ] are matrices of the same order, say m × n, then A + B = B + A.
Associative Law: For any three matrices A = [ aij ], B = [ bij ], C = [ cij ] of the same order, say m × n,( A + B) + C = A +( B + C).
Existence of additive identity: Let A = [ aij ] be an m × n matrix and O be an m × n zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition.
The existence of additive inverse: Let A = [ aij ] m × n be any matrix, then we have another matrix as – A = [– aij ] m × n such that A +(– A) =(– A) + A = O. So – A is the additive inverse of A or negative of A.