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4 it is worth noting that when we rst introduced the

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Unformatted text preview: 3 R2 · C 4 1 7 = 2 6 −4 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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