16. Systems of Linear Equations 1 Matrices and Systems of Linear ...
16. Systems of Linear Equations 1 Matrices and Systems of Linear ...
16. Systems of Linear Equations 1 Matrices and Systems of Linear ...
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March 31, 2013 16-2In that case, if A = (a ij ) <strong>and</strong> B = b jk , then C = A · B is the m × p matrixC = (c ik ) defined byn∑c ik = a ij b jk .j=1Thus the element c ik is the dot product <strong>of</strong> the ith row <strong>of</strong> A <strong>and</strong> the jthcolumn <strong>of</strong> B.Both the operations <strong>of</strong> matrix addition <strong>and</strong> matrix multiplication areassociative. That is,(A + B) + C = A + (B + C), (AB)C = A(BC).Multiplication <strong>of</strong> matrices is not always commutative, even for squarematrices. For instance, if[ ][ ]1 11 0A = <strong>and</strong> B = ,0 11 1then,AB =[2 11 1]<strong>and</strong> BA =[1 11 2].Let us consider some matrices A, B <strong>and</strong> illustrate these concepts.Example 1[ ] [ ]2 3 3 −1 2A = , B =1 2 1 2 3[ ]9 4 13C = A · B =5 3 8Example 2A =⎡⎢⎣B · A not defined[ ]2 1A T =3 22 −1 31 2 13 −2 2⎤⎥⎦ , B =⎡⎢⎣1−3−2⎤⎥⎦
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