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Subsection PMM: Properties of Matrix Multiplication

In this subsection, we collect properties of matrix multiplication and its interaction with the zero matrix (DefinitionZM), the identity matrix (DefinitionIM), matrix addition (DefinitionMA), scalar matrix multiplication (DefinitionMSM), the inner product (DefinitionIP), conjugation (TheoremMMCC), and the transpose (DefinitionTM). Whew! Here we go. These are great proofs to practice with, so try to concoct the proofs before reading them, theyll get progressively more complicated as we go.

TheoremMMZM
Matrix Multiplication and the Zero Matrix
Suppose A is an m n matrix. Then
1. A{O}_{n} = {O}_{m}
2. {O}_{p}A = {O}_{p} □

Proof Well prove (1) and leave (2) to you. Entry-by-entry, for 1 ≤ i ≤ m, 1 ≤ j ≤ p,

\eqalignno{ {\left [A{O}_{n}\right ]}_{ij} & ={ \mathop{∑ }}_{k=1}^{n}{\left [A\right ]}_{ ik}{\left [{O}_{n}\right ]}_{kj} & &\text{@(a href="#theorem.EMP")Theorem EMP@(/a)} & & & & \cr & ={ \mathop{∑ }}_{k=1}^{n}{\left [A\right ]}_{ ik}0 & &\text{@(a href="fcla-jsmath-latestli30.html#definition.ZM")Definition ZM@(/a)} & & & & \cr & ={ \mathop{∑ }}_{k=1}^{n}0 & & & & \cr & = 0 & &\text{@(a href="fcla-jsmath-latestli69.html#property.ZCN")Property ZCN@(/a)} & & & & \cr & ={ \left [{O}_{m}\right ]}_{ij} & &\text{@(a href="fcla-jsmath-latestli30.html#definition.ZM")Definition ZM@(/a)} & & & & }

So by the definition of matrix equality (DefinitionME), the matrices A{O}_{n} and {O}_{m} are equal. ■