MATHEMATICS FOR MANAGEMENT BBMP1103

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1 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Ojectives: TOPIC : MTRIX. Define mtri. ssess the clssifictions of mtrices s well s know how to perform its opertions. Clculte the determinnt for squre mtri using the cross multipliction method nd the cofctor epnsion method. Solve the liner eqution sstem using the inverse mtri method nd the Crmer s Rule. INTRODUCTION TO MTRIX. Mtri is denoted cpitl letter. On the other hnd, ever element of the mtri will e denoted smller cpitls with suscripts s shown elow: m n Where, m rows n column. Bsed on mtri ove, is mtri with the dimension of, where its elements first row first column first row second column first row third column second row first column second row second column second row third column third row first column third row second column third row third column. CLSSIFICTIONS OF MTRICES. COLUMN MTRIX (COLUMN VECTOR) ) L ) L P g e Prepred Eidin in Normn Tutor BBMP

2 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X. ROW MTRIX (ROW VECTOR) ) B ) B. ZERO MTRIX (NULL MTRIX) ) ). SQURE MTRIX ) S ) S i) DIGONL MTRIX (t lest one non-ero elements on its min digonl while ll other elements re ero) ) P ) P ii) IDENTITY MTRIX (Elements on the min digonl set to nd ll other elements set to ero) ) I ) I

3 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Eercise. () Find the vlues of tht mke digonl mtri. () Is the mtri n identit mtri? Clrif our nswer. (c) Determine the clsses for ech of the mtrices elow: i) ii) Solutions: iii).. No. Identit mtri onl eists for squre mtri.. (i) Zero Mtri (Null Mtri) - ( ) (ii) Column Mtri - ( ) (iii) Row Mtri - ( ) P g e Prepred Eidin in Normn Tutor BBMP

4 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X. MTRIX OPERTIONS. TRNSPOSE (Reverse the order of the row nd column elements so tht the rows turn into columns nd columns into rows) Emples: i) If, T ii) If B, T B. MTRIX DDITION (Onl fesile if ll the mtrices re of the sme dimension) Emples: i) ii) iii) iv) v) Cnnot e performed due to the different dimensions. MTRIX SUBTRCTION (Onl fesile if ll the mtrices re of the sme dimension)

5 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Emples: i) ii) iii) iv) Cnnot e performed due to the different dimensions. SCLR MULTIPLICTION (If mtri is multiplied sclr, then ever element of the mtri will e multiplied the sclr too) Emples: i) If ii) If B B iii) If C.... C

6 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X. MULTIPLICTION OF MTRICES (Possile onl if the numer of column of the first mtri is the sme s the numer of the row of the second mtri) Emples: Given,, B, C, I, I i) B D ii) B Is not possile due to the numer of column of mtri B is not the sme s the numer of the rows of mtri iii) C E iv) I F

7 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X v) I Is not possile due to the numer of column of mtri is not the sme s the numer of the rows of mtri I vi) I G vii) I Is not possile due to the numer of column of mtri I is not the sme s the numer of the rows of mtri viii) I I i) known s Idempotent Mtri

8 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Eercise. () Suppose:, B, C, Find: (i) (ii) B D (iii) C D (iv) B (v) C (vi) BD (vii) T () Determine mtri tht stisfies the following eqution P g e Prepred Eidin in Normn Tutor BBMP

9 MTHEMTICS FOR MNGEMENT BBMP P g e 9 Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Solutions. i) 9 ii) + 9 iii) iv) Is not possile due to the numer of column of mtri is not the sme s the numer of the rows of mtri B v) 9 vi)

10 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X vii) T 9 ) 9

11 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP. DETERMINNT. Onl for squre mtrices. Used to find to the solution to the liner eqution sstem. Denoted nd hs sclr vlue. Methods in otining the determinnt: ) Cross Multipliction Method ) Co-Fctor Epnsion Method. CROSS MULTIPLICTION METHOD ) Emple: If, B i) ii) B ) Step (Write down the first nd second columns on the right side of the mtri ) Step (dd up the multipliction products of the elements tht lie on the digonl elow) P g e Prepred Eidin in Normn Tutor BBMP

12 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Step (dd up the multipliction products of the elements tht lie on the opposite digonl elow) Step (The determinnt cn e otined sutrcting Step nd Step ). CO-FCTOR EXPNSION METHOD i) Minor of Element (The minor of the element is the determinnt of the su mtri left fter omitting the row nd column). P g e Prepred Eidin in Normn Tutor BBMP

13 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Minor ii) Cofctor of Element (The cofctor of the element is the minor of the element multiplied i j ) minor of element minor of element minor of element minor of element minor of element minor of element minor of element minor of element minor of element Cofctor P g e Prepred Eidin in Normn Tutor BBMP

14 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP iii) Determinnt ) Step (Select one row or column to perform. Generll we choose the row or column with mn eros. More esier to solve the opertion) ) Step (Perform cofctor epnsion on the selected row or column multipling the ech element in the row or column with its cofctor) c) Step (Clculte the cofctor epnsion in step in Step to determine the determinnt of mtri ) Determinnt (The sme vlue of determinnt will e otined lthough cofctor epnsion is performed on different row or column) iv) djoint Mtri (Trnspose cofctor mtri. The outcome is n djoint mtri) djoint Cofctor T P g e Prepred Eidin in Normn Tutor BBMP

15 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Eercise.. Find the determinnt for the mtrices elow: ) ) 9 c) d) e) f). Determine the determinnt for the mtrices elow using the cofctor epnsion method. ) ). ) Clculte the vlue for given the determinnt for is c) Find the determinnt for?

16 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Solutions.. ) ) 9 9 c) d) e) f). )

17 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X ). ) is ) Determinnt is not eist ecuse it is not squre mtri.

18 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X. INVERSE MTRIX. Onl for squre mtrices. Not ll mtrices hve n inverse especill when the determinnt. mtri without n inverse is known s singulr mtri. When mtri is multiplied its inverse, the following properties re then true: ) I ) I. The formul for deriving is s elow, djoin Emples I I

19 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Eercise. () () (c) Find the inverse (if there eist n) for the following mtrices. Then prove tht our nswers re correct. (i) (ii) (iii) (iv) Given: (i) Find (ii) Show tht Suppose: B c d (i) Determine B (ii) Stte the properties required for the eistence of (iii) Verif BB B B I B P g e 9 Prepred Eidin in Normn Tutor BBMP

20 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Solutions. () Find the inverse (if there eist n) for the following mtrices. Then prove tht our nswers re correct. (i) Step : Find the determinnt Step : Find djoin mtri Step : Find inverse mtri djoin (ii) Step : Find the determinnt

21 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Find minor mtri

22 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Minor Step : Find cofctor mtri Step : Find djoin mtri T Cofctor Step : Find inverse mtri djoin (iii) Step : Find the determinnt

23 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Does not eist (iv) Step : Find the determinnt 9 Step : Find minor mtri

24 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Minor Step : Find cofctor mtri Step : Find djoin mtri T Cofctor Step : Find inverse mtri djoin

25 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X () Given: (i) Find Step : Find the determinnt Step : Find djoin mtri Step : Find inverse mtri djoin

26 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X (ii) Show tht Step : Find the determinnt Step : Find djoin mtri Step : Find inverse mtri djoin

27 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP (c) Suppose: B c d (i) Determine B B c d Step : Find the determinnt B d d c c Step : Find djoin mtri c d d c Step : Find inverse mtri d d c c djoin (ii) Stte the properties required for the eistence of B d c (iii) Verif BB B B I d c d d c c d c d c c I d P g e Prepred Eidin in Normn Tutor BBMP

28 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP. SOLUTIONS FOR LINER EQUTION SYSTEM USING MTRICES. Liner eqution sstem hs to e epressed in the form of mtri eqution sstem.. Two methods will e used to solve liner eqution: () Mtri Inverse Method () Crmer s Rule Emples: TWO LINER EQUTION SYSTEM Mtri Inverse Method: Step : Convert the equtions to mtri Step : Find the determinnt of Mtri Step : Find the Mtri Minor of,,, Minor Step : Find the Mtri Cofctor of P g e Prepred Eidin in Normn Tutor BBMP

29 MTHEMTICS FOR MNGEMENT BBMP P g e 9 Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Find the Mtri djoin of T Cofctor Step : Find the Inverse of djoin Step : Solve the equtions, Emples: THREE LINER EQUTION SYSTEM Mtri Inverse Method: Step : Convert the equtions to mtri

30 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Find the determinnt of Mtri Step : Find the Mtri Minor of

31 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Minor Step : Find the Mtri Cofctor of Step : Find the Mtri djoin of T Cofctor Step : Find the Inverse of djoin

32 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Solve the equtions,, Emples: TWO LINER EQUTION SYSTEM Crmer s Rule Step : Convert the equtions to mtri Step : Find,,

33 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Find the vlue of nd Emples: THREE LINER EQUTION SYSTEM Crmer s Rule: Step : Convert the equtions to mtri Step : Find,,,

34 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X,, Step : Find the vlue of nd,,

35 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Eercise. () Epress the following liner eqution sstems in the form of mtri equtions. Susequentl, solve the equtions using the Mtri Inverse Method. (i) (ii) () (c) Solve () using the Crmer s Rule. Solve the following eqution sstem using pproprite method: Where: (i),,,, (ii) P g e Prepred Eidin in Normn Tutor BBMP

36 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Solutions. () Epress the following liner eqution sstems in the form of mtri equtions. Susequentl, solve the equtions using the Mtri Inverse Method. (i) Step : Convert the equtions to mtri Step : Find the determinnt of Mtri Step : Find the Mtri Minor of,,, Minor Step : Find the Mtri Cofctor of Step : Find the Mtri djoin of T Cofctor P g e Prepred Eidin in Normn Tutor BBMP

37 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Find the Inverse of djoin Step : Solve the equtions, (ii) Step : Convert the equtions to mtri Step : Find the determinnt of Mtri

38 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Find the Mtri Minor of Minor

39 MTHEMTICS FOR MNGEMENT BBMP P g e 9 Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Find the Mtri Cofctor of Step : Find the Mtri djoin of T Cofctor Step : Find the Inverse of djoin Step : Solve the equtions,, () Solve () using the Crmer s Rule. (i)

40 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X Step : Convert the equtions to mtri Step : Find,, Step : Find the vlue of nd (ii) Step : Convert the equtions to mtri Step : Find,,,

41 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X,, 9 Step : Find the vlue of nd

42 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X,, (c) Solve the following eqution sstem using pproprite method: Where: (i),, Step : Convert the equtions to mtri Step : Find,,, 9

43 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X,, 9 Step : Find the vlue of nd,,

44 MTHEMTICS FOR MNGEMENT BBMP P g e Prepred Eidin in Normn Tutor BBMP T o p i c M T R I X (ii),, Step : Convert the equtions to mtri Step : Find,,, 9,, 9

45 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP 9 9 Step : Find the vlue of nd,, P g e Prepred Eidin in Normn Tutor BBMP

46 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Eercise. (Pst Yers Emintion Questions). Given:, B 9 9 Find: () B () B (c) B (d) B T (Midsem Jn 9). Solve the following simultneous equtions using the Crmer s Rule Method: (Midsem Jn 9). Solve the following simultneous equtions using Inverse Method: () Write the sstem into mtri form, nd let e the coefficient mtri. () Find the determinnt of (c) Find the minor of (d) Find the cofctor of (e) Find the djoin of (f) Find the inverse of (g) Solve for,, (Midsem Jn 9). Dieri dn B. Dptkn jwpn gi soln erikut. Tunjukkn jln kerj ng lengkp. () T B () B (Finl Jn ) P g e Prepred Eidin in Normn Tutor BBMP

47 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP. Dieri sistem persmn liner erikut: () Tulis semul sistem di ts dlm entuk persmn mtriks X B () Dieri, cri nili dengn menggunkn perturn Crmer (Finl Jn ). Solve the following simultneous equtions using Inverse Method: () Write the sstem into mtri form, nd let e the coefficient mtri. () Find the determinnt of (c) Find the minor of (d) Find the cofctor of (e) Find the djoin of (f) Find the inverse of (g) Solve for,, (Finl Sept ). Given:, B Find: () B () B (c) B T (Midsem Sept ). Solve the following simultneous equtions using the Crmer s rule method: 9. Solve the following simultneous equtions using Inverse Method: P g e Prepred Eidin in Normn Tutor BBMP

48 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP () Write the sstem into mtri form, nd let e the coefficient mtri. () Find the determinnt of (c) Find the minor of (d) Find the cofctor of (e) Find the djoin of (f) Find the inverse of (g) Solve for,, (Finl Sept ) GOOD LUCK SELMT MENCUB P g e Prepred Eidin in Normn Tutor BBMP

49 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP P g e 9 Prepred Eidin in Normn Tutor BBMP