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Fig. 36
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Notice that the subscript used with each element denotes rst the ROW and then the COLUMN in which the element appears. This convention, of giving rst the row and then the column, is always used. For example, the notation a23 denotes the element at the intersection of the second row and the third column (the notation a23 can be read as a, two, three ). When it is deemed necessary, the row and column numbers can be separated by a comma, as, for example, a16;11 . Problem 24 (a) How many elements in a seventh-order determinant (b) Using a with subscript, identify the element at the intersection of the fth row and third column of a determinant of order ve or higher. (c) What is the di erence between a1;11 and a11;1
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The Second-Order Determinant
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By de nition, a second-order determinant has two rows and two columns, and thus four elements. Using the standard notation of section 3.1, the general form of the second-order determinant is shown in Fig. 37.
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CHAPTER 3 Determinants and Equations
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Fig. 37
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The second-order determinant is the basic determinant; later on we ll nd that all determinants, regardless of order, can be expressed in terms of second-order determinants. So far we ve not given any meaning to the set of symbols in Fig. 37; that is, we ve not de ned what a determinant is to mean. For reasons that will become apparent to us later on, the value, D, of a second-order determinant is now de ned to be as follows, The value of a second-order determinant is de ned as being equal to the product a11 a22 minus the product a12 a21 . Thus, by de nition, Fig. 37 has the value 36
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In eq. (36) we used the two arrows to show the two multiplications, but such arrows are not, of course, normally shown. Note that there is no proof of anything required here, because we are simply de ning what a second-order determinant is. Later on you ll nd out why it s convenient to de ne the meaning in this way. We should also mention that there are no restrictions on what the numbers, represented by the letters, can be. Thus the elements of a determinant can be real numbers or complex numbers, or any combination of such numbers.
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  4 3   Find the value of the determinant  2 5
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Solution This is a second-order determinant, and thus by eq. (36) we have, 4 5 3 2 20 6 14; answer:
Find the values of the second-order determinants in problems 25 through 29. Problem 25   6 2  (a)  4 2 Problem 26     4 2   1     3 6 2  1 3 4  2   2
  1   4   3   3
 2   1 
CHAPTER 3 Determinants and Equations
Problem 27    5 10    6 4   4 4   7 5 Problem 28    5x y     y 2xy  Problem 29    x 2 x 5     4 2 x 
Minors and Cofactors. Value of any Nth-order Determinant
Let N be the order of any determinant, where N is any whole number greater than 1. Thus, if N 2 we have a second-order determinant, if N 3 we have a third-order determinant, and so on. An Nth-order determinant refers to a determinant of any order whatever, and is just the algebraic way of saying that we are talking about a determinant of any order in general. Every element of a determinant has what is called the minor of that element. To nd the MINOR of any element, strike out the row and column in which the element is located. The determinant that remains is de ned to be the minor of that element. This is illustrated in the following example. Example
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