CHPTER d. MTRICES University of Bhrin Deprtment of Civil nd rch. Engineering CEG -Numericl Methods in Civil Engineering Deprtment of Civil Engineering University of Bhrin Every squre mtrix hs ssocited with it sclr clled determinnt. There re different methods to find the determinnt of squre mtrix. mong these methods, is the widely used method of finding the determinnt with expnsion by cofctors. Slide No.
Nottion Given squre mtrix, the determinnt of this mtrix is denoted by either det() or For exmple if then det ( ) 9 9 Slide No. Nottion 9 ( ) det Note tht represents mtrix, rectngulr rry, n entity unto itself, while det() represents sclr, number ssocited with the mtrix. The difference is only in the form. 9 Slide No.
Determinnt of mtrix The determinnt of mtrix [] is the sclr. Exmple The determinnt of the mtrix [] is nd the determinnt of the mtrix [-.] is. Slide No. Determinnt of mtrix By definition, the determinnt of mtrix is given by c b d d bc Slide No.
Exmple: mtrix Find det() if det ( ) d bc ()() ()() - - Slide No. Method for Finding the Determinnt of Higher-order Mtrices Expnsion by Cofctor Definition: Given mtrix, minor is the determinnt of ny squre submtrix of Tht is, given squre mtrix, minor is the determinnt formed by by removl of n equl number of rows nd columns Slide No. 9
Expnsion by Cofctor Exmples: Minors If 9 then nd 9 re both minors since Slide No. nd re both squre submtrices of, while nd 9 re not minors since 9 9 is not submtrix of nd [ ], lthough submtrix of, is not squre. Slide No.
Expnsion by Cofctor Definition: Given mtrix [ ij ], the cofctor of the element ij is sclr obtined by multiplying together the term (-) ij nd the minor obtined from by removing the i th row nd j th column In other words, to compute the cofctor of the element ij we first form submtrix of by crossing out both the row nd column in which the element ij ppers. Then we find the determinnt of the submtrix nd finlly multiply it by the number (-) ij Slide No. Exmple : Cofctor Find the cofctor of the element in the following mtrix 9 We first note tht ppers in the (,) position. The submtrix obtined by crossing out the second row nd first column is Slide No.
Exmple (cont d): Cofctor det 9 9 9 (9) () 9 Since ppers in the (,) position, i, nd j. Thus, (-) ij (-) (-) - Therefore, the cofctor of (-) (-) Slide No. Exmple : Cofctor Find the cofctor of the element 9 in the following mtrix 9 We first note tht 9 ppers in the (,) position. The submtrix obtined by crossing out the third row nd third column is Slide No.
Exmple (cont d): Cofctor det 9 () () Since 9 ppers in the (,) position, i, nd j. Thus, (-) ij (-) (-) Therefore, the cofctor of 9 () (-) - Slide No. Expnsion by Cofctors To find the determinnt of squre mtrix of rbitrry order:. Pick ny one row or ny one column of the mtrix.. For ech element in the row or column selected, find its cofctor.. Multiply ech element in the row or column selected by its cofctor nd sum the results.. This sum is the determinnt of. Slide No.
Exmple : Mtrix Find the determinnt of the following mtrix: Expnding by the first row, the determinnt cn be evluted s follows: Slide No. Exmple (cont d): Mtrix det ( ) ( ) Cofctor of Cofctor of Cofctor of Slide No. 9 9
det ( ) Exmple (cont d): Mtrix Therefore, ( ) [ ] [ ] [ ] Slide No. Exmple : by Mtrix Find det() if First, check to see which row or column contins the most zeros nd expnd by it. Thus expnding by the second column gives Slide No.
Slide No. Exmple (cont d): by Mtrix ( ) ( ) ( ) ( ) ( ) ) (cofctor of cofctor of cofctor of cofctor of Slide No. Exmple (cont d): by Mtrix ( ) ( ) ( ) [ ] [ ] [ ] [ ] [ ] [ ] () () 9 ) ()( ()() ) ()( ()() ()() ()()
Exmple (cont d): by Mtrix ( ( ) ( ) ( ) [ )() ()()] [()() ()( ) ] [ ] [ ] Slide No. Exmple (cont d): by Mtrix Therefore, ( cofctor of ) ( cofctor of ) ( cofctor of ) ( ) ( ) ( ) ( ) 9 (cofctor of ) Slide No.
Properties of. If the elements of ny two rows (columns) re equl, the determinnt equls zero. det( ) Slide No. Properties of. If the vlues in ny row (column) re proportionl to the corresponding vlues in nother row (column), the determinnt equls zero. det[] becuse colu m n colu m n or the first colu mn is proportionl to the third colu mn Slide No.
Properties of. If ll the elements in ny row (column) equl zero, the determinnt equl zero. 9 det( ) Slide No. Properties of. If mtrix B is obtined from mtrix by multiplying every element in one row (one column) of by constnt c, then B c. det, B () ( B) [ () () ] () Slide No. 9
Properties of. The vlue of the determinnt is not chnged by dding ny row (column) multiplied by constnt c to nother row (column)., () () M ultiplying the second row by (-) nd dding it to the first Row produces the following mtrix B B, B () ( )() Slide No. Properties of. If ny two rows (columns) re interchnged, the sign of the determinnt will be chnged. () () nd () () Slide No.
Properties of. For n n n nd ny constnt c, the det(c) c n det(). or () (), () () [ () () ], [ 9() () ] Slide No. Properties of. The determinnt of squre mtrix equls tht of its trnspose, tht is, T T,, () () B () () Slide No.
Properties of 9. If squre mtrix is plced in the digonl form using property, then the product of the elements on the digonl equls the determinnt of, () () M ultiplying the first row by / nd dding itto the sec Ro w produces mtrix with zero ele ment in the secon Row nd first colu mn s follows: Slide No. Then, multiplying the second row by / nd dding it to the first row results in the following digonl mtrix: Therefore, the determinnt of is Slide No.
Properties of.if mtrix hs zero determinnt, then is sid to be singulr mtrix, tht is, the inverse of does not exist. Slide No. Rnk of Mtrix Definition The rnk of mtrix, designted r(), is the order of the lrgest nonzero minor of. Slide No.
Rnk of Mtrix Exmple : Rnk of Mtrix Find the rnk of The lrgest minor tht cn be formed from is of order. There is only one such minor, nmely det(), nd it is zero. Thus, the rnk of will be or less. Checking ll the 9 minors of order, we find tht ech of them is lso equl to zero. Slide No. Rnk of Mtrix Exmple (cont d): Rnk of Mtrix i.e., i.e., () ( ) ( ) () Hence, the rnk of will be or zero. Checking minors of order, we find tht one which is not zero (in fct ll re nonzero); therefore, r ( ) Slide No. 9 9
Rnk of Mtrix Exmple : Rnk of Mtrix Find the rnk of ll minors of order equl zero, so the rnk of will be or less. Checking ll minors of order, we find one of them, nmely ( ) () differs from zero, so r() Slide No. Inverse of Mtrix by Cofctor nd djoint Mtrices Definition The cofctor mtrix ssocited with n n n mtrix is n n n mtrix c obtined from by replcing ech element of by its cofctor. Slide No.
Slide No. Inverse of Mtrix by Cofctor nd djoint Mtrices Exmple: Cofctor Mtrix Wht is the cofctor mtrix of, if Slide No. Inverse of Mtrix by Cofctor nd djoint Mtrices Exmple (cont d): Cofctor Mtrix ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( c
Inverse of Mtrix by Cofctor nd djoint Mtrices Definition The djoint of n n n mtrix is the trnspose of the cofctor mtrix of. If the djoint of is denoted by, then ( c ) T Slide No. Inverse of Mtrix by Cofctor nd djoint Mtrices Exmple: djoint of Mtrix Find for mtrix given the previous exmple. From previous exmple Therefore, c c T ( ) Slide No.
Inverse of Mtrix by Cofctor nd djoint Mtrices Theorem If, then the inverse of my be obtined by dividing the djoint of by the determinnt of, tht is Slide No. Inverse of Mtrix by Cofctor nd djoint Mtrices det ( ) c ( ) ( ) Exmple: Inverse of Mtrix Find the inverse of the following mtrix: () () () ( ) () () ( ) () c T ( ) Slide No.