Grade 12 - Exp. / Using matrices 32 exercises (100% corrected)

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1. System of linear equations without calculator E.6173 We wish to solve the following sys-tem : ( S ) : x + 2 y + z = 8 x + 2 z = 7 x + 3 y = 7 1 a Perform the following matrix product : 1 2 1 1 0 2 1 3 0 · x y z b What can be said about the solutions of the following matrix equation : 1 2 1 1 0 2 1 3 0 · x y z = 8 7 7 2 a Perform the following matrix product : − 6 3 4 2 − 1 − 1 3 − 1 − 2 · 1 2 1 1 0 2 1 3 0 b Using the previous question, simplify the following equation : - 6 3 4 2 -1 -1 3 -1 -2 · 1 2 1 1 0 2 1 3 0 · x y z = -6 3 4 2 -1 -1 3 -1 -2 · 8 7 7 c Give the triplet ( x ; y ; z ) solution of the ( S ) system of equations. E.6183 Consider the two matrices A and B defined by: A = 1 1 − 2 0 − 3 2 − 2 0 3 ; B = 9 3 4 4 1 2 6 2 3 1 Perform the matrix product : A · B 2 Consider the system of linear equations with three un-knowns : ( S ): x + y − 2 z = 1 − 3 y + 2 z = − 1 − 2 x + 3 z = − 1 a Justify that the system ( S ) admits as solution set that of the matrix equation : A · x y z = 1 − 1 − 1 b Deduce the set of solutions of the system ( S ) . E.6184 Consider the two matrices A and B : A = 2 1 − 2 1 − 1 1 3 − 1 0 ; B = 1 2 − 1 3 6 − 4 2 5 − 3 1 Carry out product : A · B 2 Consider the following system of three equations with three unknowns : ( S ): x + 2 y − z = 5 3 x + 6 y − 4 z = 13 2 x + 5 y − 3 z = 11 Determine the set of solution triplets of ( S ) . E.6096 Consider the matrix: A = 4 1 3 2 1 a Calculate the matrix 6 · A − A 2 . b Deduce that A is invertible and that its inverse matrix, denoted A − 1 , can be written as : A − 1 = ¸ · I + ˛ · A , où ¸ and ˛ are two real numbers to be determined. c Establish that the matrix A − 1 admits as expression : A − 1 = 2 5 − 1 5 − 3 5 4 5 2 Propose a resolution of the system : 4 x + y = 4 3 x + 2 y = − 1 2. System of linear equations with calculator E.6195 Consider the ( S ) system of equations defined by: ( S ) : x + 2 y + 3 z = 7 − x − 2 z = − 4 x + y + 2 z = 5 1 Using the calculator, give the inverse matrix of the ma-trix M où : M = 1 2 3 − 1 0 − 2 1 1 2 2 a Consider the matrix X defined by: X = x y z . Give the two matrices A and B such that the matrix equation A · X = B has the same solutions as the system https://chingmath.fr chapExoCorrec/6173 sacados/6173 chapExoCorrec/6183 sacados/6183 chapExoCorrec/6184 sacados/6184 chapExoCorrec/6096 sacados/6096 chapExoCorrec/6195 sacados/6195
( S ) . b Give the set of solutions of the system ( S ) . E.6185 1 Using the calculator, determine the inverse of the matrix A defined by: A = − 1 1 − 2 3 − 2 3 3 − 1 − 1 2 Solve the following system : ( S ) : − x + y − 2 z = 1 3 x − 2 y + 3 z = − 4 3 x − y − z = − 5 E.6186 1 Using the calculator, determine the inverse of the matrix A defined by: A = 2 1 − 2 1 − 1 − 2 1 1 − 1 2 Solve the following system : 2 x + y − 2 z = 1 x − y − 2 z = 1 x + y − z = 0 E.5113 Consider the square matrix A of or-der 3 defined by: A = 1 − 2 − 2 − 1 1 2 0 2 − 1 1 Using matrix calculation software, establish that the ma-trix A is invertible and give the inverse matrix of the matrix A . 2 Solve the following system of equations : ( S ) : x − 2 y − 2 z = 11 − x + y + 2 z = − 9 2 y − z = − 3 E.5114 Solve the system of equations below using a computer program and the matrices : ( S ) : 3 x − 3 y − 2 z = 0 3 x + y + z = − 2 − 2 x + 3 y + 2 z = − 1 E.8635 Solve the system of equations below using a computer program and the matrices : ( S ) : 2 x − 3 y + z = − 8 − 3 x + 2 y − 3 z = 3 x − y + z = − 1 E.8636 Solve the system of equations below using a computer program and the matrices : ( S ) : − x − 3 y − 2 z = − 1 − 3 x − 2 y + 2 z = 0 3 x + 3 y − z = − 1 E.8637 Solve the system of equations below using a computer program and the matrices : ( S ) : − 2 x − 3 y − 2 z = 2 − 3 x − 3 y + 2 z = 1 x + 2 y + 3 z = − 2 E.6208 During a marketing campaign, the company B distributes a pen or key ring; it costs the com-pany 0.80 e per pen and 1.20 e per key ring distributed. At the end of the day the company distributed 550 objects and it cost 540 e . We are looking for the number s of pens and the number c of key rings distributed. 1 Write a system translating this situation. 2 Show that the previous system is equivalent to : R · X = T où R = 1 1 0.8 1.2 and X and T are matrices to be specified. 3 Solve the system using the calculator. Interpret the re-sult. 3. System of linear equations and modeling E.6207 The company U supplies its customers with refills for water coolers and has the following past results : Nombre de recharges en milliers 1 3 5 Total annual cost de production in hundreds of euros 11 27.4 83 Total production cost is modeled by a function C defined for any real number x in the interval 0 ; 10 by: C ( x ) = a · x 3 + b · x 2 + c · x + 10 où a , b and c are real numbers. Where x denotes the number of thousands of refills produced, C ( x ) is the total production cost in hundreds of euros. Justify that the triplet ( a ; b ; c ) is a solution of the system ( S ) . ( S ) : a + b + c = 1 27 a + 9 b + 3 c = 17.4 125 a + 25 b + 5 c = 73 https://chingmath.fr chapExoCorrec/6185 sacados/6185 chapExoCorrec/6186 sacados/6186 chapExoCorrec/5113 sacados/5113 chapExoCorrec/5114 sacados/5114 chapExoCorrec/8635 sacados/8635 chapExoCorrec/8636 sacados/8636 chapExoCorrec/8637 sacados/8637 chapExoCorrec/6208 sacados/6208 chapExoCorrec/6207 sacados/6207
E.6215 The company U supplies its customers with refills for water coolers and has the following past results : Nombre de recharges en milliers 1 3 5 Total annual cost de production in hundreds of euros 11 27.4 83 Total production cost is modeled by a function C defined for any real number x in the interval 0 ; 10 by: C ( x ) = a · x 3 + b · x 2 + c · x + 10 où a , b and c are real numbers. Where x denotes the number of thousands of refills produced, C ( x ) is the total production cost in hundreds of euros. The triplet ( a ; b ; c ) is assumed to be a solution of the system ( S ) . ( S ) : a + b + c = 1 27 a + 9 b + 3 c = 17.4 125 a + 25 b + 5 c = 73 On pose : X = a b c 1 a Write this system as M · X = Y où M and Y are matrices to be specified. b Assume that the matrix M is invertible. Using the cal-culator, determine the triplet ( a ; b ; c ) solution of the system ( S ) . 2 Using this modeling, what would be the total annual pro-duction cost for 8 000 refills of water produced? E.6110 The company U supplies its customers with refills for water coolers and has the following past results : Nombre de recharges en milliers 1 3 5 Total annual cost of produc-tion in hundreds of euros 11 27.4 83 the total cost of production is modeled by a function C de-fined for any real number x in the interval 0 ; 10 by: C ( x ) = a · x 3 + b · x 2 + c · x + 10 où a , b , c real numbers. Where the number x denotes the number of thousands of re-fills produced, C ( x ) is the total production cost in hundreds of euros. We admit that the triplet ( a ; b ; c ) is a solution of the system ( S ) : a + b + c = 1 27 a + 9 b + 3 c = 17.4 125 a + 25 b + 5 c = 73 We pose : X = a b c . 1 a Write this system as M · X = Y où M and Y are ma-trices to be specified. b Assume that the matrix M is invertible. Using the cal-culator, determine the triplet ( a ; b ; c ) solution of the system ( S ) . 2 Using this modeling, what would be the total annual pro-duction cost for 8 000 refills of water produced? https://chingmath.fr chapExoCorrec/6215 sacados/6215 chapExoCorrec/6110 sacados/6110
02468101214161820246810IJK D1D2D3M1M2M3N1N2P1P2 E.6233 A leisure park offers its visi-tors acrobranch courses. The various courses are modeled by the Γ graph below où whose vertices correspond to the five trees marking their ends. Each path is represented by an edge of the graph and can be taken in either direction. 1 a Give, without justification, the degree of each of the vertices (the answer may be presented in tabular form où the vertices will be put in numerical order) . b Give the matrix M associated with the graph (vertices will be put in numerical order) . 2 Which of the graphs below represents a subgraph of the starting graph : a b c You will justify your answer only in the case où you are not in the presence of a subgraph. 3 A giant slide is installed on the tree 4 . The shape of this slide is modeled by a function f whose curve C is given below in an orthonormal reference frame. This curve passes through the points I , J and K of co-ordinates (2 ; 8.1) , respectively, (10 ; 2.5) et (20 ; 0) . The function f is defined on 0 ; 20 by: f ( x ) = ax 2 + bx + c où a , b and c are three real numbers. a Justify that a , b and c are solutions of the system : 400 a + 20 b + c = 0 100 a + 10 b + c = 2.5 4 a + 2 b + c = 8.1 b Determine the matrices X and V so that the previous system is equivalent to : U · X = V où U = 400 20 1 100 10 1 4 2 1 c Determine a , b and c . E.6216 Consider the function f defined by the third-degree polynomial: f ( x ) = a · x 3 + b · x 2 + c · x + 2 où a , b , c ∈ R The function f admits the following images : f ( − 1) = − 8 ; f (1) = 8 ; f (2) = 28 1 Write a system ( S ) of three equations and three un-knowns whose triplet ( a ; b ; c ) is a solution. 2 Note X the column matrix a b c . a Express two matrices A and Y such that the equation ( E ) matrix below admits the same triplet solution of the system ( S ) of equations : ( E ) : A · X = Y . b Using the calculator, give the approximate value of the triplet solution of ( S ) to the nearest hundredth. 4. Graphs and paths of lengths p E.6156 The graph below represents the main towns in the D 1 , D 2 and D 3 departments and the roads link-ing them. https://chingmath.fr chapExoCorrec/6233 sacados/6233 02468101214161820246810IJK chapExoCorrec/6216 sacados/6216 chapExoCorrec/6156 sacados/6156 D1D2D3M1M2M3N1N2P1P2
Lundi 12/09S. 38Lundi 19/09S. 39Lundi 26/09S. 40AFDAFDAFD DA1.Départ3.Rochepercée5.Picrouge7.Colvert9.Cascadedesanglais2.Passerelle4.Coldes3vents6.Refuge8.PontNapoléon10.Arrivée ABCDEFT 1 a How many paths connect the city M 1 to the city P 1 . b How many paths connect the city M 3 to the city P 2 . 2 a Give the matrix A = a ij où the coefficient a ij rep-resents ˇ the number of roads connecting city M i with city N j ı for i ∈ [[1;3]] and j ∈ [[1;2]] . b Give the matrix B = b ij où the coefficient b ij repre-sents ˇ the number of roads connecting city N i with city P j ı for i ∈ [[1;2]] and j ∈ [[1;2]] . 3 a Give the matrix C = c ij of dimension 3 × 2 obtained by the product of the matrix A by the matrix B . b Give the values of the coefficients c 11 and c 32 . 4 What conjecture can be made by comparing the results of questions 1 and 3 ? E.6164 Three friends Abondance, Fortune and Désirée (they will be considered in this order throughout the exercise) exchange their make-up products at the begin-ning of each week. The graph below shows the different ex-change possibilities between two weeks : 1 a How many exchange possibilities allow Abondance to find its make-up products at the beginning of the week 40 ? b How many exchange possibilities allow Fortune to have, in week 40 , Abondance’s make-up products in week 38 . 2 a We note M the matrix representing exchanges be-tween week 38 and week 39 . More precisely, the coeffi-cient m ij represents the number of possible exchanges in the first week between the i ième friends and the j ième friends. b We note N the matrix representing exchanges during the second week. More precisely, the coefficient n ij represents the number of exchanges during the second week between the i ième friends and the j ième friends. 3 We note P = p ij the matrix, square of order 3 , product of the matrix M by the matrix N : a Give the matrix C . b Give the values of coefficients p 11 and p 22 . 4 Comparing the results of questions 1 and 3 b , what conjecture can be made? E.6341 A mountain hiking guide de-scribes the possible routes around a rocky peak. The description of the routes is given by the graph below. The vertices of this graph correspond to remarkable places. The edges of this graph represent the possible paths between these places. 1 Give a route from D to A passing through all the ver-tices of the graph once but not necessarily taking all the paths. 2 Is there a route from D to A using all the paths once? Justify your answer. 3 Let M be the adjacency matrix associated with this graph, the vertices being taken in alphabetical order. We give M 5 . M 5 = 56 78 75 82 59 57 54 40 26 31 78 88 95 89 96 57 50 65 48 30 75 95 68 68 77 68 46 73 52 23 82 89 68 62 98 49 29 79 67 13 59 96 77 98 50 82 80 40 24 46 57 57 68 49 82 36 25 68 49 16 54 50 46 29 80 25 10 73 60 5 40 65 73 79 40 68 73 32 14 48 26 48 52 67 24 49 60 14 6 39 31 30 23 13 46 16 5 48 39 2 a What does the number 89 located in the second row and fourth column represent? b Determine the number of routes from D to A using 5 trails. Name one such route passing through Red Peak. E.6340 The graph below shows all the taxiways used by aircraft at a given airport. These taxi-ways, on which aircraft taxi before or after landing, are called taxiways . The edges of the graph represent the traffic lanes (the ˇtaxi-waysı) and the vertices of the graph are the intersections. 1 Write the matrix M associated with this graph (arrange vertices in alphabetical order) . 2 Name all paths of length 3 connecting A to T . https://chingmath.fr chapExoCorrec/6164 sacados/6164 Lundi 12/09S. 38Lundi 19/09S. 39Lundi 26/09S. 40AFDAFDAFD chapExoCorrec/6341 sacados/6341 Extrait d'Antilles-Guyane Juin 2013 DA1.Départ3.Rochepercée5.Picrouge7.Colvert9.Cascadedesanglais2.Passerelle4.Coldes3vents6.Refuge8.PontNapoléon10.Arrivée chapExoCorrec/6340 sacados/6340 ABCDEFT
ABCDEFG ABCDEFT ABCD ABCDEF ABCDEFGHI E.6342 In the graph below, the ver-tices represent different residential or activity areas within a municipality. An edge connecting two of these vertices indi-cates the existence of a main access route between two corre-sponding locations. 1 Give the matrix M associated with the graph (the ver-tices will be put in alphabetical order) . 2 The matrix is given : M 3 = 2 7 8 5 5 5 3 7 8 12 13 12 8 5 8 12 12 15 13 13 5 5 13 15 12 13 12 8 5 12 13 13 10 12 5 5 8 13 12 12 8 7 3 5 5 8 5 7 2 Determine, with justification, the number of paths of length 3 connecting A and F , then list them. E.6111 The graph below shows all the taxiways used by aircraft at a given airport. These taxi-ways, on which aircraft taxi before or after landing, are called taxiways . The edges of the graph represent the (the ˇtaxiwaysı) taxiways and the vertices of the graph are the intersections. This graph is oriented to indicate the direction of traffic for aircraft in the different lanes : 1 Write the matrix M associated with this graph (arrange vertices in alphabetical order) . 2 Using the calculator, give the value of the coefficient (1 ; 7) of M 3 . 3 Name all paths of length 3 connecting A to T . E.6241 Consider the graph opposite. 1 Give the adjacency matrix M of this graph. (we’ll consider the vertices of the graph using alphabetical order) . 2 Using the calculator, give the expression of the matrix M 4 . E.6240 In the graph opposite, we’ve indicated, for the same city, the direction of traffic flow for vehicles on the different avenues. 1 Can we find a route of any length that allows us to go from D to B respecting the direction of traffic? Justify your answer. 2 Write the matrix M associated with this graph (we’ll arrange the vertices in alphabetical order) . 3 Using the calculator, give the expression of the matrix M 3 . E.8318 Consider the graph G below : 1 Give the matrix M associated with the graph G (vertices will be arranged in alphabetical order) . 2 We give : M 3 = 6 11 9 10 3 3 4 9 3 11 8 10 9 8 8 3 4 3 9 10 6 5 3 3 8 9 3 10 9 5 6 7 7 3 3 1 3 8 3 7 2 2 1 3 0 3 8 3 7 2 2 1 3 0 4 3 8 3 1 1 4 7 5 9 4 9 3 3 3 7 6 6 3 3 3 1 0 0 5 6 2 a Give the number of paths of length 3 connecting vertex B to vertex D . b Give the number of paths of length 4 connecting vertex F to vertex I . 5. Jointly defined sequences E.5950 We define the two sequences u n and v n by the relations : u 0 =0 ; v 0 =1 ; u n +1 = u n + v n 2 v n +1 = u n + 2 v n 3 for all n ∈ N https://chingmath.fr chapExoCorrec/6342 sacados/6342 ABCDEFG chapExoCorrec/6111 sacados/6111 ABCDEFT chapExoCorrec/6241 sacados/6241 ABCD chapExoCorrec/6240 sacados/6240 ABCDEF chapExoCorrec/8318 sacados/8318 ABCDEFGHI chapExoCorrec/5950 sacados/5950
1 Determine the values of the terms u 2 and v 2 . 2 Consider the matrix M : M = 1 2 1 2 1 3 2 3 a Check, for any n ∈ N : u n +1 v n +1 = M · u n v n b Establish, using reasoning by recurrence, the identity for any n ∈ N : u n v n = M n · 0 1 c Using the calculator, give the values of the terms u 10 and v 10 . E.5357 Consider the two real sequences a n and b n defined by: a n +1 = 0.7 · a n + 0.6 · b n b n +1 = 0.3 · a n + 0.4 · b n For any natural integer n , we define the matrix-column U n defined by: U n = a n b n 1 Determine the matrix T realizing for any natural number n the following equality: U n +1 = T · U n 2 Assume : a 0 = 2 3 and b 0 = 1 3 . a Determine the values of a 1 and b 1 using the matrix relationship established in question 1 . b What conjecture can be made about the behavior of the sequences a n and b n ? 3 It is assumed that : a 0 =0.3 and b 0 =0.7 . a Demonstrate, using reasoning by recurrence, that for any natural number n , we have the relation: U n = T n · U 0 b Using a calculator or formal calculation software, de-termine the expression of the matrices U 2 , U 3 and U 5 to the nearest 10 − 6 . c What can we conjecture about the behavior of the suites a n and b n ? E.6093 Every young parent uses just one brand of baby food every month. Three brands X , Y , Z share the market. Let n be a natural number. Let: X n the event : ˇ the brand X is used in the month n ı Y n the event : ˇ the mark Y is used the month n ı Z n the event : ˇ the mark Z is used the month n ı Event probabilities X n , Y n , Z n are denoted x n , y n , z n , re-spectively. Each brand’s advertising campaign changes the breakdown : A buyer of the brand X the month n , has the following month : 50 % chance of remaining loyal to this brand ; 40 % chance of buying the brand Y ; 10 % chance of buying brand Z . A buyer of brand Y the month n , has the following month : 30 % chance of remaining loyal to this brand ; 50 % chance of buying the brand X ; 20 % chance of buying brand Z . A buyer of brand Z the month n , has the following month : 70 % chance of remaining loyal to this brand ; 10 % chance of buying the brand X ; 20 % chance of buying brand Y . 1 Represent the transition graph associated with this phe-nomenon of evolutions. 2 Noting U n the column matrix x n y n z n , determine the ma-trix expression T realizing the relationship : U n +1 = T · U n 3 This phenomenon of evolutions is assumed to be initial-ized with the values : x 0 = 0.3 ; y 0 = 0.5 ; z 0 = 0.2 Using a matrix and manual calculation, determine the percentage of buyers devolved to each of these brands in the second month? 6. Diagonalizable matrix and sequences E.5944 We define the two sequences u n and v n by the relations : u 0 = 0 v 0 = 1 ; u n +1 = u n + v n 2 v n +1 = u n + 2 v n 3 for any n ∈ N We define the matrix sequence X n by the relation X n = u n v n Consider the matrix A defined by: 1 2 1 2 1 3 2 3 1 a Demonstrate that for any natural number n , we have : X n +1 = A · X n b Demonstrate by recurrence the following relation for any natural number n : X n = A n · X 0 2 We define the matrices P and P : https://chingmath.fr chapExoCorrec/5357 sacados/5357 chapExoCorrec/6093 sacados/6093 chapExoCorrec/5944 sacados/5944
P = 4 5 6 5 − 6 5 6 5 ; P = 1 2 − 1 2 1 2 1 3 a Show that the matrices P and P are invertible. b Determine the matrix B diagonal realizing the follow-ing equality: P · B · P = A c Demonstrate by recurrence that for any natural num-ber n : A n = P · B n · P 3 a Give the expression of the matrix B n as a function of n . b Establish that the matrix X n admits for expression, for any natural number n : X n = 3 5 − 3 5 · 1 6 n 3 5 + 2 5 · 1 6 n 4 Deduce the limits of the sequences u n and v n . 7. Unclassified financial years E.8824 Consider the Fibonacci sequence F n defined by: u 0 =0 ; u 1 =1 ; F n +1 = F n + F n − 1 for all n ∈ N ∗ Let U n be the row matrix F n F n +1 . 1 Give the square matrix A of order 2 with real coefficients that satisfies : U n +1 = U n · A 2 Consider the two matrices : D = 1+ 5 2 0 0 1 − 5 2 P = 1 1 1+ 5 2 1 − 5 2 a Show that the matrix P is invertible, then determine its inverse matrix. b Establish that : A = P · D · P − 1 c Deduce an expression for the term F n in terms of n . Binet’s formula : (1843) For any natural number n , the term F n of rank n in the Fibonacci sequence has the value : F n = 1 + 5 n − 1 − 5 n 2 n · 5 https://chingmath.fr chapExoCorrec/8824 sacados/8824