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Stitz-Zeager_College_Algebra_e-book

# In the expansion a11 det a11 a12 det a12 11n

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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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