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**Unformatted text preview: **or all m × n
matrices A and B scalars k ,
k (A + B ) = kA + kB
• Zero Product Property: If A is an m × n matrix and k is a scalar, then
kA = 0m×n if and only if k = 0 or A = 0m×n As with the other results in this section, Theorem 8.4 can be proved using the deﬁnitions of scalar
multiplication and matrix addition. For example, to prove that k (A + B ) = kA + kB for a scalar k
and m × n matrices A and B , we start by adding A and B , then multiplying by k and seeing how
that compares with the sum of kA and kB .
k (A + B ) = k [aij ]m×n + [bij ]m×n = k [aij + bij ]m×n = [k (aij + bij )]m×n = [kaij + kbij ]m×n
As for kA + kB , we have
kA + kB = k [aij ]m×n + k [bij ]m×n = [kaij ]m×n + [kbij ]m×n = [kaij + kbij ]m×n
which establishes the property. The remaining properties are left to the reader. The properties in
Theorems 8.3 and 8.4 establish an algebraic system that lets us treat matrices and scalars more or
less as we would real numbers and variables, as the next example...

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