733. Table Indicating the Region of Convergence of Some Important Series The region of convergence is indicated by the heavy line. BOOK VIII CHAPTER I INTRODUCTORY CHAPTER IN THE THEORY OF DETERMINANTS I. DETERMINANTS OF Two Rows Equations of the First Degree in Two Unknown Quantities 734. Determinants of Four Elements.-The expression r = a,b,—abı is written in various ways, for example and is read the determinant of the elements (a, b), (a, b). The term a, b, is called the principal term of r; the horizontal rows (for example a,b) are called simply rows (numbered first, second, 2 1 etc., downward), and the vertical columns (for example) are called simply columns (numbered first, second, etc., from left to right). 735. The Principle of Development of r.-The determinant r is equal to the difference between the products of the elements in the two diagonals, in which the principal term is +. Accordingly REMARK. Every term of a determinant of four elements contains one clement from each row and each column. 5. Find a determinant which is 0 if a b Develop the determinants: = c: d. 736. The Elimination of One Unknown Quantity from Two Equations of the First Degree. in which it is assumed that the coefficients a1 and a3 of the unknown quantity are not 0. It is desired to know what the necessary and sufficient conditions are that the same value of x satisfies both equations. Multiply equation (1) by a, and equation (2) by -a, and add, equations (3) and (4). Equation equations (1) and (2) are not con Equations (1) and (2) lead to (4) is a necessary condition in case tradictory. Equation (4) is also a sufficient condition; since (2) can be derived from (1) and (4), thus from (4) substituting in (1), xb, whence a,b,, equation (2). Q. E. D. a2b1 Similarly equation (1) can be derived from (2) and (4). 737. The Eliminant. ris called the eliminant of the system of equations (1) and (2), or of the equivalent system y (1') and (2′) are found by putting x = in equation (1) and (2) of 2736. Hence, according to what precedes, the eliminant of two equations of the first degree, (1) and (2), is the determinant of the coefficients of the unknown quantity and the constant terms, or of the coefficients of the unknown quantities in case the given equations are written in the form of homogeneous equations of the first degree. The resultant is found by placing the eliminant equal to 0. REMARK. The results can be described as follows: If the equations (1) and (2) are compatible, or what amounts to the same thing, in case these equations are satisfied by the same values of x, then the identity (3) exists among the coefficients of these equations. 738. The Solution of two Equations of the First Degree in the Case when the Determinant of the Unknown Numbers is not 0. Let the equations be Multiply equation (1) by b, and equation (2) by -b, and add; then we get (3) (4) 1 Multiply equation (1) by -a, and equation (2) by a, and add, and get 2 RULE. The denominator of x and y is the determinant of the coefficients of the unknown quantities of equations (1) and (2); the numerator of x is found by substituting in the denominator the terms c1 and C2 in the second members of (1) and (2) for the coefficients a and a, of x; similarly the numerator of y is found by substituting C1 and c for the coefficients b, and b, of y. The correctness of the solutions in (5) can be verified thus: Substitute in (1) and (2) the values of x and y in (5), and they will become or arranging the numerators with respect to the c's, That is, the values in (5) satisfy equations (1) and (2). 1. With two Unknown Quantities.-If c, c, 0, then equations (1) and (2), 2738, are homogeneous and give x = 0, y = 0 as solution of the equations when a,b,- ab1 = 0. 2. With Three Unknown Quantities.-Substitute in equations (1) and (2), 738, Y x= y= Z' and they become Then, instead of equations (5), we obtain the equations Y, Z are proportional to the determinants of the table found by striking out the first, then the second, and finally the third column. |
