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Problem 33* Find the value of the determinant   2 2   1 0    3 1   1 0
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* See footnote given with the solution to problem 33 (after Appendix).
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CHAPTER 3 Determinants and Equations
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Problem 34 Find the value of the determinant  1  6   4  0 3 1 2 0 2 3 0 0  5  2   5  5 
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Some Important Properties of Determinants
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There are some very useful properties of determinants that we should be familiar with. It won t be necessary that we stop, here, to give a formal proof of each property, but we ll explain the meaning of each one in detail. To begin, let us state that the properties we ll study in this section are true for determinants of any order. Furthermore, any statement we will make about rows will also apply to columns, and vice versa. We ll also, when necessary, make use of the fact that the value D of a determinant is the same regardless of which rows or columns we happen to make use of in the process of nding the value of D; this was pointed out in section 3.3. As we found in section 3.3, the value D of a determinant is equal to the sum of a number of terms, each term being a product of several di erent elements of the determinant. The rst property of determinants (property 1) that we now wish to consider concerns the number and nature of such terms. We ll proceed in steps, as follows. As already de ned, an Nth-order determinant is a square array of N 2 elements arranged in an equal number of N rows and N columns. A determinant is equal to a single value, which we ll generally denote by D. In section 3.2 we learned that the second-order determinant is the basic or prototype determinant, its structure and value being de ned by eq. (36). In section 3.3 we learned that any Nth-order determinant can be expanded in terms of the minor determinants of any row or column. Each such minor determinant can then be expanded in terms of its minor determinants, until nally, continuing on in this way, the original Nth-order determinant will be found to be equal to a sum of a number of SECOND-ORDER determinants, the value of each such second-order determinant being found by eq. (36). With this in mind, let us see what we can discover about the number and the nature of the terms whose sum equals the value D of the determinant. To do this, let us begin with the third-order determinant N 3 , shown in Fig. 39.
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Fig. 39
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CHAPTER 3 Determinants and Equations
Now expand Fig. 39 in terms of, say, the elements of the rst row; thus          a22 a23     a12  a21 a23  a13  a21 a22   D3 a11    a a a32 a33 a33 a32  31 31 and hence, by the basic eq. (36) we have that D3 a11 a22 a33 a11 a23 a32 a12 a21 a33 a12 a23 a31 a13 a21 a32 a13 a22 a31 38
Inspection of eq. (38) shows that the value of a third-order determinant is equal to the sum of 6 terms, each term being the product of 3 elements. Note that each term has as factors one element, and only one, from each row and each column. For example, in the term a12 a23 a31 , element a12 is in row 1 and column 2; element a23 is in row 2 and column 3; element a31 is in row 3 and column 1; that is, every row and every column is represented once, and only once, in every term, as inspection of the subscripts in eq. (38) will show. To continue, let us next consider a fourth-order determinant N 4 such as is shown in Fig. 40.
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