...Therefore D = — D, that is, 2 D = 0, or D = 0. aad Thus, 6 6 8 cc/ = ab/ + aec + übe — aec — a6/ — dbc = 0. Theorem 4. If all the elements of a row (or...be simplified by applying this theorem. Thus, - 6 82 -3 -1 15 - 20 = 2-5. 3 -4 = 2.5.3.4. = 480. 4 -1 1 -1 903 904 A COLLEGE ALGEBRA 905 906 907 Corollary....

...228.) Similarly for interchange of two columns according to the final theorem of Art. 229. 4. If all of the elements of a row {or column) are multiplied by the same factor m, the determinant is multiplied by m. PROOF. Of the elements which are multiplied by m, one...

...determinant is equal to zero. For every term of the expansion contains a factor from the row of zeros. THEOREM 4. If all the elements of a row (or column) are multiplied by a constant k, '' \ determinant is multiplied by k. For every term of the exp. -1 sion is multiplied...

...transformation and evaluation of determinants. They lead up to a simple proof of Theorem 2. THEOREM 3. If all the elements of a row (or column) are multiplied by the same quantity, the value of the determinant is multiplied by this quantity. For, each term of a determinant...

...interchange rows and columns. The sign of D is changed if two contiguous rows or columns are interchanged. If all the elements of a row or column are multiplied by a constant K, the new determinant obtained has the value KD. The infinite determinant 0 1 1 ± 1 0...

...interchange rows and columns. The sign of D is changed if two contiguous rows or columns are interchanged. If all the elements of a row or column are multiplied by a constant K, the new determinant obtained has the value KD. The infinite determinant 0 1 1 ± 1 0...

...of lower order. Multiplying each row by the reciprocals of the items in the first columns, we have (If all the elements of a row (or column) are multiplied by the same number n, the determinant is multiplied by n.) RECIPROCAL OP COLUMN 1 COLUMN 1 1 1 0.54 0.49 0.41 X 1.000...

...same manner it can be proved that if two rows or columns are identical or proportional, .4=0; that if all the elements of a row or column are multiplied by k, the value of the new determinant is kA ; that columns and rows may be interchanged, keeping the...

...on evaluating D, using first column elements we get D,= flil fl.,, «; 31 fl12 a22 a32 Property 2. If all the elements of a row (or column) are multiplied by a constant k, then the determinant so obtained is k times of the original determinant. ie, '11 "12...