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CHAPTER 11 Matrix Algebra. Networks
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Consider next the general matrix of m rows and n columns, that is, the general m by n matrix. The standard notation associated with the general m n matrix is shown in Fig. 273.
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In the above, note that aij denotes the general element of the matrix at the intersection of any ith row and jth column ( eye-th row and jay-th column). With the foregoing in mind, we re now free to de ne some of the operations of matrix algebra. We begin as follows, where it should be noted that the plural of matrix is matrices ( MAY trah seez ). Let A and B denote two matrices. We de ne that two such matrices can be equal, that is, A B, only if (a) A and B have the same number of m rows and the same number of n columns; that is, only if both are m n matrices, and (b) all corresponding elements of A and B are equal. For example, if A and B are both 2 3 matrices, thus, A a11 a21 a12 a22 a13 a23 ! and B b11 b21 b12 b22 b13 b23 !
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then A B only if a11 b11 , a12 b12 , . . . ; a23 b23 . Next, two matrices can be added or subtracted only if they have the same number of m rows and the same number of n columns. If this requirement is met, then we de ne that the sum or di erence of two matrices is obtained by adding or subtracting corresponding pairs of elements in the two matrices. For example, the sum or di erence of the two 2 3
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matrices given above is equal to " a11 b11 A B a21 b21 " a11 b11 A B a21 b21
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a13 b13 a23 b23 a13 b13 a23 b23 # #
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a12 b12 a22 b22 a12 b12 a22 b22
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Since, for example, a11 b11 b11 a11 , it follows, from inspection of the above, that the addition and subtraction of matrices can be done in any order we wish that is, matrix addition and subtraction are commutative operations; thus and A B B A A B B A
Suppose, now, that the above two matrices happened to be EQUAL matrices, A B. By the de nition of equality already laid down, this means that a11 b11 , a12 b12 , and so on. Therefore, if A B, the sum of A and B at the top of the page becomes ! 2a11 2a12 2a13 A B 2A 2a21 2a22 2a23 Likewise, if we were dealing with the sum of say three equal matrices, then all the above 2 coe cients would be 3 coe cients, and so on, for the sum of any number of equal matrices. Therefore, to be consistent, we must de ne that a constant times a matrix is obtained by multiplying EVERY ELEMENT of the matrix by the constant. Thus, if k is any constant, and A is (for example) a 2 3 matrix, then k times A is equal to
Fig. 274
and so on, in the same way, for the product of k and any matrix. (Note that this rule is di erent from that for determinants, in which a constant k multiplies only the elements of any one row or any one column of the determinant.) Problem 218 elements arranged in A 5 9 matrix is a rectangular array of rows and columns. The notation a4;6 denotes the elefour and six. A unit ment at the intersection of matrix. If A is a 6 5 matrix, and if A B, matrix is always a matrix. then B is a Problem 219 If A is a 3 4 matrix and B is a 4 3 matrix, does the sum A B exist Problem 220 If A 3 1 2 7 4 5 ! and B ! ; then; A B
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