Matrices 13: determinant properties and rules continued

Transcription

1 Mtrices : determinnt properties nd rules continued nthony Rossiter Deprtment of utomtic Control nd Systems Engineering

2 Introduction Previous videos introduced the concepts of determinnt but it ws cler tht in generl these would be rther tedious to compute. This video introduces rules nd shortcuts which llow much esier computtion. For mtrix with coefficients ij nd cofctors ij, determinnt is defined from expnsion long ny row or column, tht is: n j1 ij ij n i1 ij ij

3 Reminder of video 1. For upper or lower tringulr nd digonl mtrices, the determinnt is the product of the digonl elements. 2. If mtrix hs n entire row or column of zeros, the determinnt is zero. 3. Scling ny row by λ results in scling of the determinnt by λ. 4. Multiplying n nxn mtrix by sclr λ modifies the determinnt by λ n. In this video we develop properties which identify when determinnt might be zero thus sving unnecessry computtion.

4 IF 2 ROWS/COLUMNS RE IDENTICL THEN THE DETERMINNT IS ZERO

5 If two rows re the sme, the (2x2) We begin with 2x2 exmple. determinnt is zero. b ; b b 0 b In this cse the result flls out directly from the definition. It is cler the sme result follows if two columns re the sme.

6 Exploiting results from 2x2 determinnts If two rows re the sme, the (3x3) determinnt is zero. Without loss of generlity, mke row 2 equl to row ; cof ( ) Use the determinnt definition long row

7 If two rows re the sme, the (3x3) determinnt is zero. Here we give n lterntive proof for 3x3 exmple ; cof ( ) In this cse the cofctors for row 2 must mtch those for row 1, but with opposite signs. Use the determinnt definition long row 1 nd row ( ) ( ) ( ) Clerly must be ZERO!!

8 If two rows re the sme, the determinnt of 4x4 is zero. This follows directly from the result for 3x3. Without loss of generlity we will illustrte with row 4. 3,1 4,1 3,2 4,2 3,3 4,3 3,4 4,4 It is cler tht every cofctor for the 4 th row is mde up of 3x3 determinnt, where the 3x3 determinnt hs common rows. Hence use result of previous slide Extension to common columns etc is obvious.

9 MTLB exmple Row 2 = row 1 col 2 = col 3

10 Corollry If row (or column) is multiple of nother row (or column) then the determinnt is zero. This follows directly from the rule tht: 1. Scling ny row by λ results in scling of the determinnt by λ. 2. Hence one could choose λ to scle the rows to be exctly equl which gives determinnt of ZERO; hence the originl determinnt must hve been zero.

11 MTLB exmple Row 2 is twice row 1 Column 3 is 3x column 2

12 DDING MULTIPLE OF NY ROW TO NOTHER ROW DOES NOT CHNGE THE DETERMINNT.

13 dding multiple of ny row to nother row does not chnge the determinnt. This results builds on the two erlier results. 1. If two rows re the sme, the determinnt is zero. 2. Scling ny row by λ results in scling of the determinnt by λ. This video demonstrtes the result for common rows. Extension to common columns is obvious/equivlent.

14 Define the determinnt fter dding row to nother row. Without loss of generlity we will illustrte with dding row 1 to row 2. 2,1 3,1 4,1 2,2 3,2 4,2 2,3 3,3 4,3 2,4 3,4 4,4 ; B 2,1 3,1 4,1 2,2 3,2 4,2 2,3 3,3 4,3 Extension to common columns etc is obvious. 2,4 3,4 4,4 B ( ) ( ) ( ) ( 14 ) B 14

15 Define the determinnt fter dding row to nother row. Rerrnge the determinnt clcultion bck to n underlying mtrix. B C C 2,1 3,1 4,1 2,2 3,2 4,2 2,3 3,3 4,3 2,4 3,4 4,4 ; C 3,1 4,1 3,2 4,2 3,3 4,3 3,4 4,4 Clerly det(c)=0 s it hs 2 common rows.

16 MTLB exmples dd row 1 to row 2 dd col 4 to col 3

17 Define the determinnt fter dding multiple of row to nother row. Without loss of generlity we will illustrte by dding λ x row 1 to row 2. B B B 2,1 3,1 4,1 ( 2,2 3,2 4,2 C 2,3 3,3 4,3 ) 2,4 3,4 4,4 ; ( B 2,1 3,1 4,1 ) ( 2,2 14 3,2 4,2 ) 2,3 3,3 4,3 ( Extension to common columns etc is obvious. 2,4 14 3,4 4,4 )

18 MTLB exmples dd 0.4 row 2 to row 3 Subtrct 0.5 col 3 from col 1

19 Find the determinnt (using properties) B

20 Find the determinnt (using properties)

21 Summry of rules 1. For upper or lower tringulr nd digonl mtrices, the determinnt is the product of the digonl elements. 2. If mtrix hs n entire row or column of zeros, the determinnt is zero. 3. Scling ny row by λ results in scling of the determinnt by λ. 4. Multiplying n nxn mtrix by sclr λ modifies the determinnt by λ n. 5. dding multiple of ny row to nother row does not chnge the determinnt. 6. If two rows (or two columns) re equl then the determinnt is zero. 7. If multiple of row (col) is equl to nother row (col) then the determinnt is zero.