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Unformatted text preview: + B ) + C = A + (B + C )
• Identity Property: If 0m×n is the m × n matrix whose entries are all 0, then 0m×n is
called the m × n additive identity and for all m × n matrices A
A + 0m×n = 0m×n + A = A
• 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 = 0m×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 = [aij ]m×n , we are looking
for a matrix B = [bij ]m×n so that A + B = 0m×n . By the deﬁnition of matrix addition, we must
have that aij + bij = 0 for all i and j . Solving, we get bij = −aij . 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 = [−aij ]m×n . The long and short of this is: to get t...
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