Matrices and Determinants

Transcription

1 Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner equtions. Importnt Vocbulry Define ech term or concept. Entry of mtri One of the rel numbers tht mkes up mtri. Order of mtri Indictes the number of rows nd columns of mtri. A mtri hving m rows nd n columns is sid to be of order m n. Squre mtri A mtri in which the number of rows nd the number of columns is equl. Min digonl For squre mtri, ll entries,, in which i = j. ij Row mtri A mtri tht hs only one row. Column mtri A mtri tht hs only one column. Elementry row opertions A set of opertions tht cn be performed on n ugmented mtri of given system of liner equtions tht produce new ugmented mtri corresponding to new (but equivlent) system of liner equtions. Guss-Jordn elimintion The process of reducing mtri to reduced row-echelon form. I. Mtrices (Pges ) If m nd n re positive integers, n m n mtri is... How to write mtrices nd identify their orders An m n mtri hs m rows nd n columns. An ugmented mtri is... mtri derived from system of liner equtions, ech written in stndrd form with the constnt term on the right side. 141

2 14 Chpter 8 Mtrices nd Determinnts A coefficient mtri is... mtri derived from the coefficients of system of liner equtions (but not including the constnt terms). Emple 1: Consider the following system of equtions. + y z = 5 3y + z = y = 1 () Write the ugmented mtri for this system. (b) Wht is the order of the ugmented mtri? (c) Write the coefficient mtri for this system. (d) Wht is the order of the coefficient mtri? II. Elementry Row Opertions (Pges ) The elementry row opertions on mtri re: How to perform elementry row opertions on mtrices Two mtrices re row-equivlent if... one cn be obtined from the other by sequence of elementry row opertions. A mtri in row-echelon form hs the following three properties: 1. All rows consisting entirely of zeros occur t the bottom of the mtri.. For ech row tht does not consist entirely of zeros, the first nonzero entry is 1 (clled leding 1). 3. For two successive (nonzero) rows, the leding 1 in the higher row is frther to the left thn the leding 1 in the lower row.

3 Section 8.1 Mtrices nd Systems of Equtions 143 A mtri in row-echelon form is in reduced row-echelon form if... every column tht hs leding 1 hs zeros in every position bove nd below its leding 1. III. Gussin Elimintion with Bck-Substitution (Pges ) To solve system of liner equtions using Gussin Elimintion with Bck-Substitution,... How to use mtrices nd Gussin elimintion to solve systems of liner equtions If, during the elimintion process, you obtin row with zeros ecept for the lst entry, you cn conclude tht the system hs no solution. Emple : Solve the following system using Gussin Elimintion with Bck-Substitution. + y + z = 1 + y + 3z = 1 3y + 5z = 11 The solution is = 0, y =, nd z = 1.

4 144 Chpter 8 Mtrices nd Determinnts IV. Guss-Jordn Elimintion (Pges ) Emple 3: Apply Guss-Jordn elimintion to the following mtri to obtin the unique reduced row-echelon form of the mtri How to use mtrices nd Guss-Jordn elimintion to solve systems of liner equtions : : : Emple 4: Solve the following system using Guss-Jordn elimintion. y + 3z = 1 + y 4z = 6 + 3y z = 13 The solution is =, y = 4, nd z = 3. Homework Assignment Pge(s) Eercises

5 Section 8. Opertions with Mtrices 145 Nme Section 8. Opertions with Mtrices Objective: In this lesson you lerned how to dd nd subtrct mtrices, multiply mtrices by sclrs, nd multiply two mtrices. Importnt Vocbulry Define ech term or concept. Sclrs Sclr multiple Zero mtri Mtri multipliction Identity mtri of order n I. Equlity of Mtrices (Pge 587) Nme three wys tht mtri my be represented. 1) How to decide whether two mtrices re equl ) 3) Two mtrices re equl if they hve the sme order nd re equl. II. Mtri Addition nd Sclr Multipliction (Pges ) To dd two mtrices of the sme order,... How to dd nd subtrct mtrices nd multiply mtrices by sclrs To multiply mtri A by sclr c,...

6 146 Chpter 8 Mtrices nd Determinnts Emple 1: Let A = nd B = Find () A + B nd (b) B Let A, B, nd C be m n mtrices nd let c nd d be sclrs. Give n emple of ech of the following properties of mtri ddition nd sclr multipliction: 1) Commuttive Property of Mtri Addition: ) Associtive Property of Mtri Addition: 3) Associtive Property of Sclr Multipliction: 4) Sclr Identity Property: 5) Distributive Property (two forms): If A is n m n mtri nd O is the m n zero mtri consisting entirely of zeros, then A + O =. The dditive identity for the set of ll m n mtrices is the m n mtri. III. Mtri Multipliction (Pges ) When multiplying n m n mtri A by n n p mtri B, to obtin the entry in the ith row nd jth column of AB,.... How to multiply two mtrices Emple : If A is 3 5 mtri nd B is 6 3 mtri, find the order, if possible, of the product () AB, nd (b) BA.

7 Section 8. Opertions with Mtrices 147 Nme Emple 3: Find the product AB, if 1 7 A = nd B = 3 List four properties of Mtri Multipliction: If A is n n n mtri, the identity mtri I of order n hs the property tht nd. IV. Applictions of Mtri Opertions (Pges ) Mtri multipliction cn be used to represent system of liner equtions. The system = b 1 = b = b cn be written s the mtri eqution, where A is the coefficient mtri of the system nd X nd B re column mtrices. 3 How to use mtri opertions to model nd solve rel-life problems

8 148 Chpter 8 Mtrices nd Determinnts Emple 4: Consider the following system of liner equtions = = = Write this system s mtri eqution AX = B, nd then use Guss-Jordn elimintion on the ugmented mtri [A : B] to solve for the mtri X. Additionl notes Homework Assignment Pge(s) Eercises

9 Section 8.3 The Inverse of Squre Mtri 149 Nme Section 8.3 The Inverse of Squre Mtri Objective: In this lesson you lerned how to find the inverses of mtrices nd use inverse mtrices to solve systems of liner equtions. Importnt Vocbulry Define ech term or concept. Inverse of mtri I. The Inverse of Mtri (Pge 60) To verify tht mtri B is the inverse of the mtri A,... How to verify tht two mtrices re inverses of ech other II. Finding Inverse Mtrices (Pges ) If mtri A hs n inverse, A is clled or nonsingulr. Otherwise, A is clled. How to use Guss-Jordn elimintion to find the inverses of mtrices To hve n inverse, mtri must be. Not ll squre mtrices hve inverses. However, if mtri does hve n inverse, tht inverse is. To find the inverse of squre mtri A of order n, Emple 1: Find the inverse of the mtri A =

10 150 Chpter 8 Mtrices nd Determinnts III. The Inverse of Mtri (Pge 606) If A is mtri given by nd only if true, the inverse of A is given by: b A =, then A is invertible if c d. Moreover, if this condition is How to use formul to find the inverses of mtrices 1 A = The denomintor is clled the of the mtri A. 3 9 Emple : Find the inverse of the mtri B =. 7 IV. Systems of Liner Equtions (Pge 607) If A is n invertible mtri, the system of liner equtions represented by AX = B hs unique solution given by. How to use inverse mtrices to solve systems of liner equtions Emple 3: Use n inverse mtri to solve (if possible) the system of liner equtions: 1 + 8y = y = 15 Homework Assignment Pge(s) Eercises

11 Section 8.4 The Determinnt of Squre Mtri 151 Nme Section 8.4 The Determinnt of Squre Mtri Objective: In this lesson you lerned how to find minors, cofctors, nd determinnts of squre mtrices. Importnt Vocbulry Define ech term or concept. Determinnt Minors Cofctors I. The Determinnt of Mtri (Pges ) The determinnt of the mtri A = 1 b1 is given by b How to find the determinnts of mtrices det(a) = A = = The determinnt of mtri of order 1 1 is defined s Emple 1: Find the determinnt of the mtri A =. 1 II. Minors nd Cofctors (Pge 613) Complete the sign ptterns for cofctors of 3 3 mtri, 4 4 mtri, nd 5 5 mtri: Sign Pttern for Cofctors 3 3 mtri 4 4 mtri 5 5 mtri How to find minors nd cofctors of squre mtrices

12 15 Chpter 8 Mtrices nd Determinnts Emple : Use the mtri A = 1 0 to find: 0 3 () the minor M 13, nd (b) the cofctor C 1. III. The Determinnt of Squre Mtri (Pges ) Applying the definition of the determinnt of squre mtri to find determinnt is clled. How to find the determinnts of squre mtrices Emple 3: Find the determinnt of the mtri: A = Emple 4: Describe strtegy for finding the determinnt of the following mtri, nd then find the determinnt of the mtri B = Homework Assignment Pge(s) Eercises

13 Section 8.5 Applictions of Mtrices nd Determinnts 153 Nme Section 8.5 Applictions of Mtrices nd Determinnts Objective: In this lesson you lerned how to use Crmer s Rule to solve systems of liner equtions nd how to use determinnts nd mtrices to model nd solve problems. I. Crmer s Rule (Pges ) Crmer s Rule sttes tht if system of n liner equtions in n vribles hs coefficient mtri A with nonzero determinnt A, the solution of the system is = A1 A A, = A,, 1 where the ith column of A i is n = A n A. How to use Crmer s Rule to solve systems of liner equtions Crmer s Rule does not pply if the determinnt of the coefficient mtri is, in which cse the system hs either no solution or. Emple 1: Use Crmer s Rule to solve the system of liner equtions. + y + z = 6 y + 3z = 1 y z = 3 II. Are of Tringle (Pge 6) The re of tringle with vertices ( 1, y 1 ), (, y ), nd ( 3, y 3 ) is Are = 1 ± where the symbol ± indictes tht the pproprite sign should be chosen to yield positive re. How to use determinnts to find the res of tringles Emple : Find the re of tringle whose vertices re ( 3, 1), (, 4), nd (5, 3).

14 154 Chpter 8 Mtrices nd Determinnts III. Lines in Plne (Pges 63 64) Three points ( 1, y 1 ), (, y ), nd ( 3, y 3 ) re colliner (lie on the sme line) if nd only if = 0. How to use determinnt to test for colliner points nd find n eqution of line pssing through two points Emple 3: Determine whether the points (, 4), (0, 3), nd (8, 1) re colliner. An eqution of the line pssing through the distinct points ( 1, y 1 ) nd (, y ) is given by = 0. Emple 4: Find n eqution of the line pssing through the points (, 9) nd (3, 1). IV. Cryptogrphy (Pges 65 67) A cryptogrm is... How to use mtrices to encode nd decode messges To use mtri multipliction to encode nd decode messges,... Homework Assignment Pge(s) Eercises