I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge
If the mtri A hs m rows nd n olumns we n write: A m m n n mn where ij represents the numer or element in the i th row nd j th olumn. Mtri Alger Pge
Speil Mtries A squre mtri hs the sme numer of rows s olumns. The min digonl of squre mtri is the digonl running from top left to ottom right. An identit mtri, denoted I, is squre mtri with ones on the min digonl nd zeros elsewhere. I The trnspose of A is otined writing rows s olumns nd olumns s rows, nd is denoted da T. Mtri Alger Pge
Equlit of Mtries If A = ( ij ) nd B = ( ij ), A = B if nd onl if ij = ij. Addition nd Sutrtion of Mtries Mtries of the sme size m e dded to nd sutrted from one nother. To do this, the orresponding elements re dded or sutrted. Mtri Alger Pge
6 7 e.g. If 9,, C B A find A + B, B + C nd B - C. A + B is not defined s A nd B re not the sme size. 6 7 B + C = 8 9 6 7 7 B C = Mtri Alger Pge
Multiplition of Mtri Numer An mtri n e multiplied numer. To do this, eh element of the mtri is multiplied tht numer. 7 e.g. If A 9 8, find A, -A. 8 *7 * * 6 A = *9 *8 * 8 6 * * *8 8 6 -A = Mtri Alger Pge 6
= Multiplition of Mtries If A is m n mtri nd B is p q mtri. For the produt AB to eist we must hve n = p, i.e. A B C m n p q m np Note tht mtri multiplition is : i. not ommuttive (i.e. AB BA). ii. ssoitive [i.e. ABC = (AB)C = A(BC)]. iii. If C = AB, the element ij is found from row i of A nd olumn j of fb, s follows: ij n i j q Mtri Alger Pge 7
6 6 7 6 8 6 7 AB 8 9 6 6 6 8 AB 6 6* * 8* i.e. Mtri Alger Pge 8
eg e.g. If A B findab &, AB. AB= Note tht when squre mtri is post- or pre-multiplied n identit mtri of the pproprite size the mtri is unhnged, i.e. AI = IA = A Mtri Alger Pge 9
DETERMINANTS, INVERSE OF A MATRIX Referene : Croft & Dvison, Chpter, Blos, Determinnt t All squre mtries, A, possess determinnt denoted : det(a), A. Determinnt of mtri If A, then det(a) = A = = d - d d A mtri whih hs zero determinnt is lled singulr. Mtri Alger Pge
Minors nd Coftors of Mtri Let ij e n element of mtri A. The minor of ij is the determinnt formed rossing out the i th row nd j th olumn of det(a). The oftor of ij = (-) i+j (minor of ij ) Note tht the term (-) i+j is lled the ple sign of the element on the i th row nd j th olumn. The following m help ou to memorize this. Mtri Alger Pge
Determinnt of Mtri Consider generl mtri, A = det(a) n e lulted l epnding long n row or olumn. For emple, epnding long the first row: A = *(its oftor) + *(its oftor) + *(its oftor) Mtri Alger Pge
e.g. Find the vlue of nd 6 * *( ) * * * * 7 7 6 Mtri Alger Pge
Alterntivel, ti l Rule of Srrus Repet the st nd nd olumn to right hnd side of rd olumn to form mtri. det(a) = Add the produt of SOLID digonls from left top to right ottom nd sutrt the produts of DASH digonls from left ottom to right top. Mtri Alger Pge
Hene *( ) *( ) ** ** *( ) * ** **( ) Mtri Alger Pge
Properties of Determinnts i. If ever element of given row (or olumn) of the squre mtri is multiplied the sme ftor, the vlue of the determinnt is multiplied tht ftor ii. If B is otined interhnged n rows (or olumns) of A, then B = - A. iii. Adding or sutrting multiple of one row (or olumn) to nother row (or olumn) leves the determinnt unhnged. iv. If A nd B re squre mtries nd tht AB eists, s, then det(ab) = det(a)det(b). v. If rows or olumns of squre mtri re equl, the determinnt of the mtri is zero. Mtri Alger Pge 6
Inverse of Mtri The inverse mtri of squre mtri A, usull denoted A -, hs the propert : AA - = A - A = I Note tht if A =, A does not hve n inverse. A, A hs n inverse Mtri Alger Pge 7
Finding the Inverse of Mtri The followings re steps to find the inverse of mtri A when A, i. Find the trnspose of A, denoted A T. ii. Reple eh element of A T its oftor. The resulting mtri is lled the djoint of A, denoted dj(a). iii. A dj ( A ) A Mtri Alger Pge 8
e g Find the inverse of & B A e.g. Find the inverse of 6 & B A det(a) = T 7 7 dj( A) 7 7 A Mtri Alger Pge 9 7
TYPES OF SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS When sstem of liner equtions is solved, there re possile outomes: i. unique solution ii. n infinite numer of solutions iii. no solution i. ( unique solution ) Mtri Alger Pge
ii. 6 6 (infinite numer of solutions) iii. ( no solution) Mtri Alger Pge
CRAMER S RULE CRAMER S RULE Referene : Croft & Dvison, Chpter, Blos, Crmer s rule is method tht uses determinnts to solve t f li ti sstem of liner equtions. i. Two equtions in unnowns q If h dd then,, tht provided Mtri Alger Pge
ii. equtions in unnowns z If where z z then,, z Mtri Alger Pge
e.g. Using Crmer s rule, solve for,. 7 9 Mtri Alger Pge
e.g. Using Crmer s rule, solve for, nd z. z z z z Mtri Alger Pge
INVERSE MATRIX METHOD INVERSE MATRIX METHOD Writing Sstem of Equtions in Mtri Form Note tht n e written s This is lled the mtri form of the simultneous equtions. q Mtri Alger Pge 6
z Similrl, z z n lso e written s z i th l t i f f t f ti i.e. the generl mtri form of sstem of equtions: AX = B where A, X nd B re mtries. AX = B Mtri Alger Pge 7
Solving Equtions Using the Inverse Mtri Method Consider the mtri form: AX = B i.e. X n e found if A - eists. A - AX = A - B I X = A - B X=A - B Mtri Alger Pge 8
e.g. Redo emple nd emple using the inverse mtri method. Mtri Alger Pge 9
GAUSSIAN ELIMINATION Referene : Croft & Dvison, Chpter, Blo Introdution Gussin Elimintion is sstemti w of simplifing sstem of equtions. A mtri, lled n ugmented mtri, whih ptures ll the properties of the equtions, is used. A sequene of elementr row opertions on this mtri eventull rings it into form nown s ehelon form (to e disussed in Pge ). From this, the solution to the originl equtions is esil found. Mtri Alger Pge
Augmented Mtri Consider the sstem of equtions, it n e represented n ugmented mtri: p g onstnts oeffiients this vertil line n e omitted s in our tetoo Mtri Alger Pge
Similrl, the following sstem of equtions: z z z n lso e written s n ugmented mtri: Mtri Alger Pge
e.g. Write down the ugmented mtries for the followings 9 7. 7. 9 7 6 z z 6 z 7. 7 6 8 z z 7 Mtri Alger Pge
e.g. Solve the sstem with the ugmented mtri: e.g. Solve the sstem with the ugmented mtri:. 7. Mtri Alger Pge
Row-Ehelon Form of n Augmented Mtri For mtri to e in row-ehelon form: i. An rows tht onsist entirel of zeros re the lst rows of the mtri. ii. For row tht is not ll zeros, the first non-zero element is one, lled leding. iii. While moving down the rows of the mtri, the leding s move progressivel to the right. Mtri Alger Pge
e.g. Determine whih of the following mtries re in rowehelon form. 78. 8. 7 Mtri Alger Pge 6
Elementr Row Opertions The elementr opertions tht hnge sstem ut leve the solution unltered re: i. Interhnge the order of the equtions. ii. Multipl or divide n eqution non-zero onstnt. iii. Add, or sutrt, multiple of one eqution to, or from, nother eqution. Mtri Alger Pge 7
Note tht t row of n ugmented mtri orresponds to n eqution of the sstem of equtions. When the ove elementr opertions re pplied to the rows of suh mtri, the do not hnge the solution of the sstem. The re lled elementr row opertions. Mtri Alger Pge 8
Gussin Elimintion to Solve Sstem of Equtions i. write down the ugmented mtri. ii. ppl elementr row opertions to get row-ehelon form. iii. solve the sstem. Mtri Alger Pge 9
e g Use Gussin Elimintion to solve 8 z e.g. Use Gussin Elimintion to solve z z The ugmented mtri is 8 The ugmented mtri is Interhnge row nd row 8 row *row 7 6 7 row *row Mtri Alger Pge
7 6 row *row Hene 6 7 7 z row /7 row / - 6 z or 7 7 z or The solution is z T T Mtri Alger Pge
eg e.g. Use Gussin elimintion to solve Mtri Alger Pge
e.g. Use Gussin elimintion to solve z z 8 z Mtri Alger Pge
Gussin Elimintion to find the Inverse of Mtri [ A I ] i. write down in form of. ii. ppl sequene of elementr row opertions to redue A to I. iii. Performing this sme sequene of elementr row opertions on I, we otin A -. Mtri Alger Pge
6 S A 7 6 Suppose A 6 6 7 8 6 6 6 7 6 6 7 8 Hene 8 A Mtri Alger Pge
EIGENVALUES AND EIGENVECTORS Referene: Croft & Dvison Chpter Blo Consider the sstem d or simpl, AX A If the sstem hs non-trivil solutions. A If the sstem hs onl the trivil solution. Mtri Alger Pge 6
Chrteristi Eqution nd Eigenvlues Consider the sstem d or AX X We see vlues of so tht the sstem hs non-trivil solutions. The sstem n e written s A I X A I The sstem hs non-trivil solutions if Mtri Alger Pge 7
A I hrteristi eqution., polnomil eqution in, is lled the The vlues of whih use the sstem to hve non-trivil solutions re lled eigenvlues. AX X Mtri Alger Pge 8
If A is n n n mtri nd X is vetor (n mtri) then there is usull no geometri reltionship etween the vetor X nd the vetor AX (figure (g elow). AX X Mtri Alger Pge 9
But if X is n eigenvetor of A then AX = X i.e. AX is slr multiple l of X. AX X Mtri Alger Pge
Suh vetors rise in the stud of free virtions, eletril sstems, hemil retions nd mehnil stress, et. m m Mtri Alger Pge
Eigenvetors Suppose tht, n eigenvlue, stisfies the sstem AX X. For eh eigenvlue there is non-trivl solution (unique up to non-zero slr multiple) of the sstem. This solution is lled n eigenvetor. Mtri Alger Pge
e.g. Find the eigenvlues nd eigenvetors of the mtri Mtri Alger A 7 det (A-I) = Hene When, 6 or 6 ( A I) ) 6 6 6 or t, t t i.e.. In generl, The eigenvetor is t Pge
When, ( I) A i.e.. In generl, ( I) A or t, t The eigenvetor is, t g Mtri Alger Pge
e.g. Find the eigenvlues nd eigenvetors of A. A A e.g.6 Find the eigenvlues nd eigenvetors of A. A A Mtri Alger Pge
APPLICATION Stiffness method is ommonl used to nlse struture t s mtri nlsis n e used to solve the prolem. For tpil em element, the reltionship etween the memer end moments nd memer end rottions is given i, qi j, qj i j L q q i j EI EI i L L EI EI i L L j j EI EI q i L L i EI i q j EI EI j L j L L Mtri Alger Pge 6
where i [ ] j - displement mtri ii ij [ ] ] ji jj - stiffness mtri q i q j [q] - fore mtri B sseml the stiffness of the whole struture, it will eome ver Struture Stiffness Mtri. B imposing pproprite oundr onditions nd end moments, the displement of the whole struture n e solved, i.e. [ ] [ K] [ Q] Mtri Alger Pge 7