Introduction to Determinnts Remrks The determinnt pplies in the cse of squre mtrices squre mtrix is nonsingulr if nd only if its determinnt not zero, hence the term determinnt Nonsingulr mtrices re sometimes lso clled regulr mtrices Unfortuntely, except for smll mtrices, or mtrices with mny zeroes, the clcultion of determinnts cn be tedious Remrks The first record of solving liner equtions is ttributed to the Bbylonins The Chinese, between 00 BC nd 00 BC, cme much closer to mtrices thn the Bbylonins The text Nine Chpters on the Mthemticl rt written during the Hn Dynsty gives the first known exmple of mtrix methods The ide of determinnt ppered in Jpn nd Europe t lmost exctly the sme time lthough Seki in Jpn certinly published first 3
Remrks Historiclly, determinnts were considered before mtrices Originlly, determinnt ws defined s property of system of liner equtions. The determinnt "determines" whether the system hs unique solution (which occurs precisely if the determinnt is non-zero). Two-by-two determinnts were considered by Crdno t the end of the 6th century nd lrger ones by Leibniz bout 00 yers lter Following him Crmer dded to the theory, treting the subject in reltion to sets of equtions 4 Determinnt is function The determinnt function ssigns sclr, det, to every squre mtrix det : nn R The domin of the determinnt function consists of ll squre mtrices nd the rnge of the determinnt function consists of ll sclrs 5 ) is mtrix Common Nottion ) is n element of mtrix 3) is the mtrix obtined from mtrix by deleting the ith row nd jth column of n n 4) det det n nn n nn 6
Common Nottion n n n n is mtrix nn is mtrix nn n n nn is determinnt 7 Determinnts of Order One The determinnt of mtrix sclr itself Exmples det 4 4 det t t is the 8 Determinnts of Order Two The determinnt of is det mtrix 5 4 Exmple det 53 4 5 8 7 3 9 3
Determinnts of Order Three 3 3 3 3 33 det 33 3 3 3 3 3 3 33 3 3 0 Using Digonls for Order Three 3 3 3 3 33 3 3 33 3 3 3 3 det 3 3 3 3 33 Remrks Using digonls is not efficient for determinnts lrger thn order three This method, when pplied to determinnts of order n is known s the Leibniz Formul 4
Prctice Find determinnt by using digonls 5 3 3 3 3 4 5 75 336 7 0 5 5 3 Using Cofctors 3 3 3 3 33 3 3 3 3 33 3 3 3 3 33 det 3 3 33 3 3 33 3 3 3 The sign of the coefficients is clculted by (-) i+j where i nd j re the row nd column numbers respectively of the coefficient. Note tht they lternte in sign. 4 Review - Submtrix 3 4 5 6 7 8 9 0 3 4 5 6 3 4 9 0 3 4 6 5 5
Determinnts of Order > i j n n det det det det n j j j j det i j j We refer to the term det s cofctor j For order 3, this becomes det det det det 3 3 6 Theorem The determinnt of n nn mtrix cn be computed by cofctor expnsion cross ny row or down ny column Row i expnsion we use det det n j i fixed i j fixed i j Column j expnsion we use det det n i j This is lso referred to s the Lplce Formul To ese computtion choose row or column with mostly zeroes 7 Determining the Sign i j sign i j even i j odd Row 3 nd Column 3 5 odd 8 6
Exmple Row expnsion 3 3 0 0 0 0 3 0 3 4 0 Column 3 expnsion 3 0 0 0 3 Column expnsion 3 0 0 0 3 0 4 7 0 9 Prctice Find determinnt by cofctor expnsion 5 5 5 5 5 5 4 4 4 4 5 5 5 4 0 5 0 0 450 5 0 0 5 0 Exmple 5 0 0 0 0 0 3 5 0 0 3 5 7 4 3 4 0 5 0 0 0 0 3 5 This result is typicl with tringulr mtrices, like the L nd U mtrices ssocited with LU decomposition Upper tringulr 0 0 0 0 0 0 0 0 0 0 Lower tringulr 0 0 0 0 0 0 0 0 0 0 7
Theorem If is tringulr mtrix, then det is the product of the min digonl enteries of det 33 nn This is lso true for digonl mtrices Exmple 3 4 5 0 3 0 0 3 5 0 0 0 4 det 3 4 4 3 Determinnt on TI 4 8
TI Exmple 5 Remrks If ll of the entries in row or column re zero, the determinnt is zero 0 3 n 0 3 n 0 3 33 3n 0 n 3 n nn This should mke sense if we consider the effect on cofctor expnsion where ll of the coefficients re zero It lso correltes with singulr mtrix since if one row or column is zero in n x n mtrix, the number of pivots is less thn n 6 Prctice Find the determinnt 8 4 6 0 6 4 0 0 3 8 6 3 44 7 9
ppliction Crmer s Rule Suppose tht is n n n invertible mtrix. Then the solution to the system of x b is given by x x xn where xi det det i th nd i is the mtrix found by replcing the i column of with b 8 Exmple 3 5 det 4 7 87 4 3 5 x 4 7 x 0 4 x 3 3 5 x det 0 7 87 87 3 4 3 5 73 x det 4 0 7 87 87 3 9 Exmple 3 5 x 4 7 x 0 4 x 3 3 3 07 x3 det 4 0 87 87 4 3 Solution is x= 73 87 07 30 0
ppliction Cross Product 3 The cross product uv is computed in R using determinnt. e e e 3 uv = u u u 3 v v v 3 0 0 where e = 0, =, 3= 0 e e 0 0 3 Exmple e e e3 uv 0 = e 0 e 0 e 3 0 3 3 0 3 0 0 0 3 = 0 6 0 0 0 = e e 6e = 6 3 u 0 v 0 3 Given mtrix We cn find Cofctor () where Co-Fctor Mtrix 3 3 3 3 33 C C C3 Cofctor C C C 3 C 3 C3 C 33 i C j M nd M is the determinnt of the submtrix obtined by removing from its i-th row nd j-th column 33
0 Find Cofctor Mtrix 4 4 Clculte the determinnt of ech submtrix, 6 noting the signs Exmple 4 0 6 8 0 6 6 4 0 Cofctor 6 6 8 4 3 Put together 34 Finding Inverse Mtrix The inverse mtrix cn be computed using the Trnspose of the Cofctor mtrix nd Determinnt of the Mtrix Cofctor T Notes: The Cofctor Mtrix is sometimes referred to s the djunct Mtrix The trnspose of the Cofctor Mtrix is referred to s the djugte Mtrix TI-89 Cofctor Function http://www.ticlc.org/rchives/files/fileinfo/33/3357.html 35 Exmple Using We cn find 0 4 4 6 4 0 Cofctor 6 6 8 4 3 8 T Cofctor 4 6 4 0 6 3 0 nd 4 6 4 6 36
Exmple 0 4 4 6 Putting it together 8 Cofctor T 4 6 4 6 0 6 3 3 4 3 3 3 0 37 3