Lecture. Iverse matrix. To be read to the music of Back To You by Brya dams DEFINITION OF INVERSE TRIX Defiitio. Let is a square matrix. Some matrix B if it exists) is said to be iverse to if B B I where I is as usual idetity matrix cosistig of zeros except diagoal elemets which all are uity. I Iverse matrix usually deoted by. So if equalities are true I exists the These equalities are used to check if a foud matrix is really the iverse.
COPUTTIONL ETHODS TO EVLUTE INVERSE TRIX There are two effective methods to evaluate the iverse matrix:. Gauss-Jorda method. Cofactors method We will provide o proof for these methods because oe ca always check the results of calculatios usig oe of the formulas or I I 3 GUSS-JORDN ETHOD Let us suppose that some osigular matrix is give. We begi with the combied matrix cosistig of matrix ad the idetity matrix I of he same size I) Usig Gauss-Jorda trasformatios applied oly to rows of combied matrix let us try to get a idetity matrix i the left side of combied matrix. t the momet whe idetity matrix I i the left side will be obtaied, the iverse matrix ca be read i the right side of combied matrix I ) 4
EXPLE OF USING GUSS-JORDN ETHOD Let a matrix is give. Write dow a combied matrix Usig Gauss-Jorda trasformatios we will try to get i the left side idetity matrix. dd to secod row a first row times ) Now add to the first row secod row times ) Idetity matrix is obtaied i the left side. This is the ed. 6 CHECKING THE INVERSE Idetity matrix is obtaied i the left side. So the iverse is stadig i the right side Check the equality I holds I ) ) ) )
EXISTENCE OF N INVERSE The process ca be fialized by idetity matrix if ad oly if the iitial matrix ca be trasformed ito echelo matrix of the full rak idetity matrix always has full rak). s rak is ot chaged i Gauss-Jorda trasformatios we ca coclude that iitial matrix should be osigular to have the iverse. 7 COFCTORS ETHOD Followig this method for ay matrix a a a a a a a a a the iverse ca be obtaied usig formula where is the determiat of the matrix. 8
9 FRO COFCTORS TO INORS Note that the order of idices of cofactors is iverse, so elemet ij c of a matrix is ot ij but ji. The cofactors ca be evaluated by usual way usig miors ji i j ji ) so a fial formula is ) ) ) ) ) ) ) ) ) This formula is especially coveiet whe the size of matrix is. EXPLE OF USING COFCTORS ETHOD Example. For a matrix cofactors are ), ), ), ), ad,. so s i the Gauss-Jorda method I ) ) ) )
EXISTENCE OF THE INVERSE REVISITED The cofactors method essetially uses the determiat of the matrix. It should be o-zero stadig i deomiator of the fractio), so the matrix should have full rak or to be o-sigular. SYSTE OF LINER EQUTION IN TRIX FOR Let us cosider agai the system of liear equatios i ukows ax ax a x b ax ax a x b ax ax a x b with o-sigular coefficiet matrix a a a a a a a a a Deotig as usual vector of ukows by x ad vector of costat terms by b the system of liear equatio ca be writte i a short form x b
TRIX ETHOD TO SOLVE SYSTES OF LINER EQUTIONS Let x b s o-sigularity of is supposed there exists a iverse matrix. ultiplyig both sides of the matrix equatio by get x b we But the product by the defiitio of iverse matrix is equal to idetity matrix I, hece x Ix x ad fially x b This is a formula expressig a solutio to the system of liear equatio. 3 EXPLE OF USING TRIX ETHOD TO SOLVE SYSTE Let the system of liear equatios is give x x 4 x x 9 The coefficiet matrix is osigular. Usig oe of the computatioal methods oe ca to fid a iverse Now a solutio to the system ca be obtaied by multiplicatio of iverse matrix by vector of costat terms x 4 4 9 x 9 ) 4 9 So the solutio is foud x, x. 4
BIBLIOGRPHY. Carl P. Simo, Lawrece Blume. athematics for Ecoomists. W.W.Norto&Compay. New-York, Lodo. 994. Chapter 7. V.Tcheriak. Lecture otes o liear algebra. Itroductory course. oscow. Dialog SU. L..