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That is, the values in (5) satisfy equations (1) and (2).

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1. With two Unknown Quantities.—If c = c = 0, then equations (1) and (2), 4738, are homogeneous and give a = 0, y = 0 as solution of the equations when a,b, - a,b, == 0.

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Hence X, −Y, Z are proportional to the determinants of the table

found by striking out the first, then the second, and finally the third column.

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and is called the determinant of the elements (a,b,c), (a,b,c,) (a,b,c,). The term a,b,c, is called the principal term of A. The definition of columns and rows given in #734 is adopted in case

of a determinant of nine elements.

Formation of the Determinant A. — Place the first and second columns to the right and next the given determinant or the first and second rows immediately below the third row of the given determinant, thus:

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2. “J”. AX \ A'. b, ^ Now form the six products of three elements which lie on lines drawn through the two diagonals and the lines parallel to them, neglecting lines which pass through but one or two elements. Place the sign + before the principal term and the remaining products of lines running from the left above toward the right below, and before the other products place the sign —.

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REMARK.—Each of the six terms of a determinant of nine elements contains an element from each column and each row.

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741. Relations Between Determinants of the Second and Third Orders.-We have

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for arbitrary values of ar, y, z, u. One can therefore write a determinant of the second order in the form of a determinant of the third order and in some cases conversely. Suppose that one uses stars (*) instead of the arbitrary elements; then one has

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0 y + =w = ary 2. 0 z

Hence, in case all the elements on the same side of the diagonal of a determinant are 0, the determinant is equal to its principal term.

Conversely, the product acyz can be written in the form of a determinant of the third order;

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742. First Property.— In order to multiply or divide a determinant of the second or the third order by m, one multiplies or divides all the elements of any row or any column by m.

Verification. —One can test any case by developing the determinants. Proof: Each term in the development of a determinant contains an element from each column or row; hence each term of the determinant expanded is multiplied by m if any column or row is multiplied by m, and therefore the determinant is multipled by m. Similarly for the case of division.

CoRoi, LARY 1. –In case all the elements of a column or a row have a common factor, then the elements of the column or row can be divided by this factor provided the determinant is multiplied by the same factor.

CoRoll,ARY 2. —In order to multiply or divide a determinant by (–1), it is sufficient to change the sign of the elements in any colunn or row.

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