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CHAPTER 3 Determinants and Equations
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Note that the second value for D3 has the same magnitude as the rst, but opposite sign. Continuing in this manner, we nd that the N! terms in the expansion of any Nth-order determinant always divide into two separate plus and minus groups, both groups having the same number of terms. If, now, any two rows, or any two columns, of the given determinant are interchanged, the same two groups of terms appear, but with opposite signs, in the new determinant. Next we have Property 4. If any two rows, or any two columns, of a determinant are identical, the value of the determinant is zero. Let D be the value of a determinant having two identical rows. Now interchange the two rows; since they are identical rows, this does not change the value of D. However, by property 3, interchanging the two rows must produce D; that is, we would have to have D D, which can only be true if D 0. Now consider Property 5. If m is a factor of every element in any row or any column, then m may be removed from the elements of that row or column and placed in front of the determinant as a multiplier of the entire determinant. Suppose that m is a factor of every element in a certain row or column of a determinant. By property 1, every term in the expansion of the determinant will contain m as a factor; but this produces the same result as rst removing m, then expanding the determinant, and then multiplying the result by m. To illustrate,      ma11 a12 a13   a11 a12 a13           ma21 a22 a23  m a21 a22 a23       ma31 a32 a33   a31 a32 a33  From this we see that property 5 gives us the rule that to multiply a determinant by a factor m, multiply every element of any one row or one column by m. Let us next take up property 6, which is stated as follows: Property 6. If every element of any row or any column is the sum of two terms, then the determinant can be written as the sum of two determinants. Using the following third-order determinant as     a11 A a12 a13   a11 a12        a21 B a22 a23   a21 a22     a31 C a32 a33   a31 a32 an example,   a13   A     a23   B   a33   C property 6 states that  a12 a13    a22 a23   a32 a33 
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Note that in the original determinant (the left-hand side of the equation) each of the elements of the rst column consists of the sum of two quantities. Each of these two quantities then appears separately as the rst columns of the two determinants on the right-hand side of the equation. The above equation, for the case of a 3rd-order determinant, can be veri ed by actually expanding the original determinant in terms of the elements of the rst column. The extension to the case of any Nth-order determinant is then apparent. Next we have our nal property, which is very useful:
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CHAPTER 3 Determinants and Equations
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Property 7. The value of a determinant is not changed if, to each element of any row, we add the corresponding element of any other row multiplied by the same number. As usual, the statement applies to columns as well as rows. To illustrate the meaning of property 7, let s begin with what we ll call the original determinant, shown on the left-hand side of eq. (40) (Fig. 41).
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