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MATRICES AND DETERMINANTS
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EXAMPLE B1 If ! 1 4 0 5 2 A B 2 7 3 0 1 ! 1 5 4 2 0 6 A B 2 0 7 1 3 1 ! 4 2 6 A B 2 6 2 6 1 ! !
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The transpose of the sum (di erence) of two matrices is the sum (di erence) of the two transposes: A B T AT BT Multiplication of Matrices The product AB, in that order, of a 1 m matrix A and an m 1 matrix B is a 1 1 matrix C  c11 , where 2 3 b11 6 7 6 b21 7 6 7 C a11 a12 a13 . . . a1m 6 b31 7 6 7 6 7 4 ... 5 bm1 " # m X a11 b11 a12 b21 . . . a1m bm1 a1k bk1
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Note that each element of the row matrix is multiplied into the corresponding element of the column matrix and then the products are summed. Usually, we identify C with the scalar c11 , treating it as an ordinary number drawn from the number eld to which the elements of A and B belong. The product AB, in that order, of the m s matrix A aij and the s n matrix B bij is the m n matrix C cij , where cij
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EXAMPLE B2 3 2 3 ! a11 b11 a12 b21 a11 b12 a12 b22 a12 b11 b12 a22 5 4 a21 b11 a22 b21 a21 b12 a22 b22 5 b21 b22 a32 a31 b11 a32 b21 a31 b12 a32 b22 3 2 32 3 2 3 5 8 I1 3I1 5I2 8I3 42 1 6 54 I2 5 4 2I1 1I2 6I3 5 4 6 7 I3 4I1 6I2 7I3 ! ! 2 6 5 8 3 7 5 2 3 0 5 6 3 9 19 0 9 4 8 2 7 4 2 2 0 4 6 2 9 46 a11 4 a21 a31 2
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Matrix A is conformable to matrix B for multiplication. In other words, the product AB is de ned, only when the number of columns of A is equal to the number of rows of B. Thus, if A is a 3 2 matrix and B is a 2 5 matrix, then the product AB is de ned, but the product BA is not de ned. If D and E are 3 3 matrices, both products DE and ED are de ned. However, it is not necessarily true that DE ED. The transpose of the product of two matrices is the product of the two transposes taken in reverse order:
MATRICES AND DETERMINANTS
[APP. B
AB T BT AT If A and B are nonsingular matrices of the same dimension, then AB is also nonsingular, with AB 1 B 1 A 1 Multiplication of a Matrix by a Scalar The product of a matrix A  aij by a scalar k is de ned by kA Ak  kaij that is, each element of A is multiplied by k. k A B kA kB Note the properties kA T kAT
k AB kA B A kB
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DETERMINANT OF A SQUARE MATRIX Attached to any n n matrix A  aij is a certain scalar function of the aij , called the determinant of This number is denoted as    a11 a12 . . . a1n     a21 a22 . . . a2n    det A or jAj or A or ... ... ... ...    a a ... a 
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