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733. Table Indicating the Region of Convergence of Some Important Series

The region of convergence is indicated by the heavy line.

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B O O K VIII

CHA PTER 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, - a,b,
is written in various ways, for example

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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, etc., downward), and the vertical columns (for example o are

i called simply columns (numbered first, second, etc., from left to fight).

| 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

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REMARK.—Every term of a determinant of four elements contains one element from each row and each column.

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5. Find a determinant which is 0 if a b = c : d. Develop the determinants:

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ac—are y—y, a 1–2, 3/1-ys *1–2, 3/1-3/s 736. The Elimination of One Unknown Quantity from Two Equations of the First Degree.

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in which it is assumed that the coefficients a, and a, of the unknown quantity are not 0. It is desired to know what the necessary and sufficient conditions are that the same value of a satisfies both equations. Multiply equation (1) by a, and equation (2) by —a, and add,

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Equations (1) and (2) lead to equations (3) and (4). Equation (4) is a necessary condition in case equations (1) and (2) are not contradictory. Equation (4) is also a sufficient condition; since (2) can be derived from (1) and (4), thus

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substituting in (1), * * = be whence a, c=b, equation (2). Q.E.D. Similarly equation (1) can be derived from (2) and (4).

737. The Eliminant. r is called the eliminant of the system of equations (1) and (2), or of the equivalent system

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(1') and (2') are found by putting a ==# in equation (1) and (2) of Ž736. 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 ac, 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

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. Multiply equation (1) by —a, and equation (2) by a, and add, and get (4) (a,b, - a,b,)y = a, c, -a,c, ; whence it follows from (3) and (4) that

RULE.—The denominator of a and y is the determinant of the coefficients of the unknown quantities of equations (1) and (2); the numerator of a is found by substituting in the denominator the terms c, and e, in the second members of (1) and (2) for the coefficients a, and a, of w; similarly the numerator of y is found by substituting c, and c, for the coefficients b, and b, of y.

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