4.3 Properties of Determinants 2

4.3 Properties of Determinants 2 - 4.3. . . .- C n c + C n...

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Unformatted text preview: 4.3. . . .- C n c + C n +1 C n +1 then we have fl fl fl fl fl fl fl fl fl fl fl 1 1 ( c 1- c ) c 1 ( c 1- c ) c n- 1 1 ( c 1- c ) 1 ( c 2- c ) c 2 ( c 2- c ) c n- 1 2 ( c 2- c ) . . . . . . . . . . . . 1 ( c n- c ) c n ( c n- c ) c n- 1 n ( c n- c ) fl fl fl fl fl fl fl fl fl fl fl = ( c 1- c )( c 2- c ) ( c n- c ) fl fl fl fl fl fl fl fl fl 1 c 1 c 2 1 c n- 1 1 1 c 2 c 2 2 c n- 1 2 . . . . . . . . . . . . 1 c n c 2 n c n- 1 n fl fl fl fl fl fl fl fl fl = ( c 1- c )( c 2- c ) ( c n- c ) fl fl fl fl fl fl fl fl fl 1 1 ( c 2- c 1 ) c 2 ( c 2- c 1 ) c n- 2 2 ( c 2- c 1 ) . . . . . . . . . . . . 1 ( c n- c 1 ) c n ( c n- c 1 ) c n- 2 n ( c n- c 1 ) fl fl fl fl fl fl fl fl fl = ( c 1- c ) ( c n- c )( c 2- c 1 ) ( c n- c 1 ) fl fl fl fl fl fl fl 1 c 2 c 2 2 c n- 2 2 . . . . . . . . . . . . 1 c n c 2 n c n- 2 n fl fl fl fl fl fl fl = = Q i<j n ( c j- c i ) 23. a) So k is the largest number of linearly independent columns of A thus rankA ( A ) = k (b) 205 PNU-MATH 4.3. Since rank ( A ) = k , there exists { a 1 ,a 2 , ,a k } a linearly independent set of columns of A So A can be written as a 11 a 1 k . . . . . . . . . . . . a n 1 a nk Since dim (row space A ) = dim (column space of A ) Therefore A = a 11 a 1 k ....
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4.3 Properties of Determinants 2 - 4.3. . . .- C n c + C n...

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