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A + B = [aij ]m×n + [bij ]m×n = [aij + bij ]m×n = [bij + aij ]m×n = [bij ]m×n + [aij ]m×n = B + A
where the second equality is the deﬁnition of A + B , the third equality holds by the commutative
law of real number addition, and the fourth equality is the deﬁnition of B + A. In other words,
matrix addition is commutative because real number addition is. A similar argument shows the
associative law of matrix addition also holds, inherited in turn from the associative law of real
number addition. Speciﬁcally, for matrices A, B , and C of the same size, (A + B )+ C = A +(B + C ).
In other words, when adding more than two matrices, it doesn’t matter how they are grouped. This
means that we can write A + B + C without parentheses and there is no ambiguity as to what this
means.3 These properties and more are summarized in the following theorem.
Theorem 8.3. Properties of Matrix Addition
• Commutative Property: For all m × n matrices, A + B = B + A
• Associative Property: For all m × n matrices, (A...
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