Yer 11 Mtrices Terminology: A single MATRIX (singulr) or Mny MATRICES (plurl) Chpter 3A Intro to Mtrices A mtrix is escribe s n orgnise rry of t. We escribe the ORDER of Mtrix (it's size) by noting how mny Rows n Columns it hs. But which is row n which is column? A row of sets goes cross n uitorium So Rows re horizontl The columns of the Prthenon stn upright n Columns re verticl So 1 2 3 4 5 6 is 3x2 mtrix, n 1 2 3 4 5 6 7 8 9 10 is 2x5 mtrix etc Sometimes we my nee to look t one specific piece of t insie the mtrix, so we refer to the positions within Mtrix s follows. Refer to the following GENERAL MATRIX:!!!"!"!"!"!!!"!"!!!"#/!"#$%! We cn, subtrct n multiply by sclr Aition A + B Simply the corresponing positions Subtrction A B Simply subtrct the corresponing positions SCALAR multipliction 2A Multiply ALL positions by the sme sclr (Why o we cll it Sclr Multipliction?) Ti-nSpire Menu > option 7 > Crete > Mtrix 1
Chpter 3B Mtrix multipliction This is ifferent to sclr multipliction, Mtrix Multipliction. You multiply the row of the first Mtrix, by ech column of the secon mtrix. Esiest expline on the bor, but here is igrm s well n you then continue by multiplying the secon row of the first mtrix, by both columns of the secon mtrix n this gives you the secon row of the resultnt mtrix. 2
Orer mtters Mtrix multipliction is NOT COMMUTATIVE A B = AB BA Also, not ll Mtrices will mrry up for multipliction. The rows n columns nee to hve certin imensions. Further, given two mtrices cn be multiplie, the resultnt mtrix will hve given orer: An m n mtrix, multiplie by n n p mtrix, will result in n m p mtrix. ** so the insie column & row imensions ispper, n you en up with the outsie row & column imensions. The rule tht shows the imensions / orer of the resultnt mtrix is; m n n p = m p ** for multipliction to be possible the insie Column & Row imensions nee to lign n = n. Mtrix Properties Distributive Distributive Sclr commuttive NOT Mtrix Commuttive A + ba = ( + b)a A = B = (A + B) (b)a = (ba) A B = AB BA You will lso nee to know bout the IDENTITY MATRIX I. (cpitl I) The integer 1 is the Ientity Element uner Multipliction. Tht is, nything multiplie by 1 is just itself J Here, uner mtrix Multipliction, the ientity mtrix oesn t chnge the originl mtrix. IA = AI = A ** note tht both I n A re Mtrices. ** " 1 0 0 0% " 1 0 0% $ ' " 1 0% $ ' I = $ ' = 0 1 0 = $ 0 1 0 0 ' etc etc # 0 1& $ ' $ 0 0 1 0' # $ 0 0 1& ' $ ' # 0 0 0 1& 3
Chpter 3C Powers of Mtrices Powers of Mtrix A! = A A A! = A A A A! = A A A A etc etc TASK: After oing the Text Book exercise questions, Consier the question: Prove A! A = AA!. As C exm question, mnully prove this by showing me n exmple. As B+ or A level exm question, prove the bove in the Generl cse. 4
Chpter 3D The Inverse Mtrix The Inverse Mtrix, is efine s the mtrix, tht when multiplie by the Originl mtrix, will hve the Ientity Mtrix s its resultnt. The inverse is nnotte with -1 in the superscript A!! " b% Given Mtrix A = $ ' then, A!! =! # c &!"!!" The term in the enomintor ( bc) is lso specil. It s clle the DETERMINANT. Remember the eterminnt (iscriminnt) in qurtics? Well in this cse, it Determines if there is n Inverse mtrix. Be creful not to confuse the mthemticl symbol for Determinnt s it is use in other res of mths for mgnitue. In mtrices, the eterminnt is enote s: A = bc Clerly if bc = 0, then we get n unefine nswer (becuse we cn t ivie by 0). Here we sy the Mtrix is SINGULAR n oes NOT hve n inverse! *** Use Technology to fin the eterminnt et will spit out -2 *** 1 2 3 4 n the clcultor We cn lso hve Mtrices in equtions: Let n unknown Mtrix be X AX = B **note** X is MATRIX J We isolte X by PRE-multiplying both sies by A!! (becuse A!! A = I n I X = X ) A!! AX = A!! B we get X = A!! B *** Remember ORDER Mtters, so you must PRE-multiply both sies. *** *** No nee to worry bout Iempotent, or Nilpotent *** 5
Chpter 3E The Trnspose Mtrix The TRANSPOSE of Mtrix The Trnspose of A (reflect the mtrix t bout the leing igonl) is A! ** Cre ifferent nottion thn the Text Book! but use MINE! ** Text Book is A but I prefer A! A = c b then, A! = c b n " b c % " g% T $ ' $ ' $ e f ' $ b e h' # $ g h i &' # $ c f i &' 6
Chpter 3F Applictions of Mtrices Mtrices n Simultneous equtions. x + by = e cx + y = f becomes c b x y = e f solve by pre-multiplying both sies by the inverse mtrix 1 bc b c e f = 1 bc e f x y = 1 bc e f You en up with n eqution tht looks like x y = g h n from there you simply sy x = g n y = h eqution solve! It looks Simpler when we isply the sitution s: If AX = B A!! A X = A!! B then, X = A!! B 7