t.i.tn#m.e.fit. multiply F.# t.ie product of flips you grid so if of arrangement is fi ) to the perspective on determinants arrangement :

Transcription

1 Determinants continued 9 Mar Here 's yet another perspective on determinants A rookanaugemeut_ ( my term ) Is a positioning n rooks on an nxn grid so that no pain threaten each other : 3 3 : tie 2 2 : # F# F# titn#mefit The rook a arrangement is fi ) to the number if ie switch two columns column flips you need to get there from FED Looks on main diagonal) Their given A we make a term for every rook arrangement : the sign multiply the rook arrangement by the all the matrix entries product

2 the G) " in the rooks ' positions theory Adding up the aforementioned thus gives det A [ Traditional expression : if you prefer det A E s9nh9 oaqpeyyyetgtniog what's the point? this is our first concrete a) an o and computational definition geometric doesn't give you a way to calculate dett for n > 2 and the cactor expansion is leaning on its uniqueness ( that is doesnt molten what rout column you expand along ) fact to show the uniqueness cactor expansion In one shows by induction that all give the above rook arrangement expression Example Find KEEL using rook arrangements sod det A (1) (5) 1907 (2) 1 4) ( 10 ) + (3) 1 4) (8) (1) (6) (8) A) 1 5) (3) (2) (6) 3

3 Except Show that when m u ep Is written out and simplified there are no occurrences a2 Solution Each rook arrangement includes at most one factor a Exercise Show from the rook deft that det A det At One feature determinants that Is 0 peat Ag det A " we use the term singular " to describe an uxu matrix A with det A O geometrically clear but Gears emphasizing is # Ais invertible *& We still don't have a fast way calculating det Al the rook defa is slow because it has n! terms

4 Determinants and elementary row operations * Rowswitdung_ flips the orientation and thus det multiplies by 1 to:] to:] µ# µ# dnteegetwaygbinwut orientation * Rowscalingn Scales the det : toy [ 841 He 5 * Rowad Doesn't alter det! " " I Is I wishes an

5 so So we can row reduce A to a diagonal matrix avoiding row scaling operations and use the following fact Proposition If A is upper triangular ( all zeros below the main diagonal ) then det A equals the product the diagonal entries Ro the only rook arrangement without a zero factor is TIT that's the only nonzero term in the rook expansion det A tea Example Find 5838 ) by row operations amazonee?ahteed so det A (1) C 3) ( 4) 3 ( Note : if we'd row sighed k times we'd have needed an extra factor ED " to compensate )

6 Here are a couple properties the det which are clear from the geometric definition Fact #& If a shape has volume 1 then its image under B has area detb and the image that was area Idet BI ldet A ) so ldettbl Idettlldetbl ten # that Buffy" AB But AB reverses orientations iff exactly one A or B does so the signs dettdetb and det AB watch too A detttttdet At Fact at A #

7 Cvaiueisruletheorem If A is an invertible nxn matrix then the unique solution I AI D Is given by I ( k Xu ) where for all i from 1 to n about the ithcolumn Aaithb outtake X INT ; dett solve example { xxtyy EY Cramer 's Rule using solution lty y ftp?gffz6 so ( x g) 41467

8 D AT D Cramer 's rule can be used to derive a formula for matrix inverses by soloing At et txez etc this Is called the cadjugate' formula for A " ; see the book if you're interested Rooraiuersrulesuppose AI D and consider m ftpgo?wn8d T sub out * for ith column Then AM fan ] Also detttx ; since the diagonal vole arrangement is the only one without zero factors dettdetm det [ at so aīf w ) : & ; detcaiibee#deta

9 B Cramer 's theorem is useful for expressing numerical stability : Example Explain Pset # 5 question 2 using Cramer 's rule solution We're Interested in the solution [ III It 1196k 93 ] 93 Ill Health which is aet 's focus on xd +9% tsu?yoo sensitive The numerator here Is very small in 5 changes ( ' 8h45] to because the following crucial numerical analysis concept : if D A where A B are much changes larger than d then small percent in A or B can lead to huge

10 62467 ' percent changes in d : tiny d Hx oranges!! the upshot : subtracting two very close numbers can lead to large percent errors in the difference If the coefficient matrix had been [ 3 to II ] we'd get detfl a to ] Ffs! 3 det[4i?hf Fyi which is now much Stabler ; no tiny numbers Graphically it's easier to use 7 you #