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Unformatted text preview: 3 R2 · C 4 1
7 = 2
14 −45 47 Note that the product BA is not deﬁned, since B is a 3 × 4 matrix while A is a 2 × 3 matrix; B has
more columns than A has rows, and so it is not possible to multiply a row of B by a column of A.
Even when the dimensions of A and B are compatible such that AB and BA are both deﬁned, the
product AB and BA aren’t necessarily equal.8 In other words, AB may not equal BA. Although
there is no commutative property of matrix multiplication in general, several other real number
properties are inherited by matrix multiplication, as illustrated in our next theorem.
Theorem 8.5. Properties of Matrix Multiplication Let A, B and C be matrices such that
all of the matrix products below are deﬁned and let k be a real number.
• Associative Property of Matrix Multiplication: (AB )C = A(BC )
• Associative Property with Scalar Multiplication: k (AB ) = (kA)B = A(kB )
• Identity Property: For a natural number k , the k × k identity matrix, denoted Ik , is
deﬁned by Ik = [dij ]k×k where
dij = 1, if i = j
0, otherwise For all m...
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