College Algebra, 2013a

8.3 Matrix Arithmetic 579
⎡ ⎤ ⎡ ⎤ ⎡
⎤ ⎡
2 3 −1 4 2+(−1) 3 + 4
⎣ 4 −1 ⎦ + ⎣ −5 −3 ⎦ = ⎣ 4+(−5) (−1) + (−3) ⎦ = ⎣
0 −7 8 1
0+8 (−7) + 1
⎤
1 7
−1 −4 ⎦
8 −6
It is worth the reader’s time to think what would have happened had we reversed the order of the
summands above. As we would expect, we arrive at the same answer. In general, A + B = B + A
for matrices A and B, provided they are the same size so that the sum is defined in the first place.
This is the commutative property of matrix addition. To see why this is true in general, we
appeal to the definition of matrix addition. Given A =[a ij ] m×n
and B =[b ij ] m×n
,
A + B =[a ij ] m×n
+[b ij ] m×n
=[a ij + b ij ] m×n
=[b ij + a ij ] m×n
=[b ij ] m×n
+[a ij ] 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 property of matrix addition also holds, inherited in turn from the associative law
of real number addition. Specifically, for matrices A, B, andC 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
ˆ Commutative Property: For all m × n matrices, A + B = B + A
ˆ Associative Property: For all m × n matrices, (A + B)+C = A +(B + C)
ˆ Identity Property: If 0 m×n is the m × n matrix whose entries are all 0, then 0 m×n is
called the m × n additive identity and for all m × n matrices A
A +0 m×n =0 m×n + A = A
ˆ Inverse Property: For every given m × n matrix A, there is a unique matrix denoted
−A called the additive inverse of A such that
A +(−A) =(−A)+A =0 m×n
The identity property is easily verified by resorting to the definition of matrix addition; just as the
number 0 is the additive identity for real numbers, the matrix comprised of all 0’s does the same
job for matrices. To establish the inverse property, given a matrix A =[a ij ] m×n
, we are looking
for a matrix B =[b ij ] m×n
so that A + B =0 m×n . By the definition of matrix addition, we must
have that a ij + b ij =0foralli and j. Solving, we get b ij = −a ij . Hence, given a matrix A,
its additive inverse, which we call −A, does exist and is unique and, moreover, is given by the
formula: −A =[−a ij ] m×n
. The long and short of this is: to get the additive inverse of a matrix,
3 A technical detail which is sadly lost on most readers.