Matrix/scalar multiplication is also analogous to vector/scalar multiplication, but matrix/matrix multiplication is somewhat more complicated. The greatest difficulty students have with matrix multiplication is that it can be a long process and therefore tedious. Since lots of arithmetic has to be done, there are ample opportunities to make simple mistakes. The process itself is straightforward. Take an
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matrix
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which represents the product of an
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matrix
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with a
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matrix
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. An entry
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is the dot product of the
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th row of
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(
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) with the
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th column of
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(
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). Using this definition iteratively for all the
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rows of
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and all the
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columns of
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will generate
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. This definition constrains the resulting dimensions of
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. Since the dot product is used, the number of columns in
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must equal the number of rows in
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so that the dotted vectors will have equal dimensions (in this case, both are equal to
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). Also,
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will have the same number of rows as
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(
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) and the same number of columns as
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(
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). This is easy to remember:
Additionally, matrix multiplication is
not commutative:
The reason is obvious when one looks back at the original definition. It is, however, both associative and distributive:
Excercise 3-2. For each of the following matrix pairs, state whether or not the multiplication is possible (do not perform the multiplication). If the multiplication is valid, state the resulting dimensions:
Valid (since
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and
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, Product Dimension:
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)
Invalid (since
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and
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)
Valid (since
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and
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, Product Dimension:
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)
Invalid (since
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and
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)