Grade 12 - Exp. / Matrix operations 61 exercises (100% corrected)

a
r641-0 1. Addition E.6809 Definition: Let m , n be two non-zero natural numbers ( m;n ∈ N ∗ ) . Con-sider two matrices A and B of dimensions m × n and note a ij and b ij their coefficients, respectively. The sum of the matrices A and B is the matrix of dimen-sions m × n noted A + B whose coefficients c ij , are defined by: c ij = a ij + b ij ∀ i ∈ [[1 ; m ]] ; ∀ j ∈ [[1 ; n ]] Perform the following matrix additions : a 1 2 − 1 0 + 2 − 3 − 1 − 2 b − 4 0 1 1 2 0 + 1 1 − 3 0 1 4 E.6155 Perform the following operations : A 1 0 2 4 1 − 1 + 0 2 1 − 1 − 1 0 b 2 3 4 1 + 5 − 2 − 1 − 1 + − 2 1 0 − 1 2. Multiplication by a real number and addition E.6154 Definition Let – be a real number ( – ∈ R ) and A a matrix of dimension m × n ( m;n ∈ N ∗ ) whose coefficients are denoted a ij We call the product of the matrix A by the scalar λ , the dimensional matrix m × n noted λ · A and whose coeffi-cients b ij are defined by: b ij = – · a ij ∀ i ∈ [[1 ; m ]] ; ∀ i ∈ [[1 ; m ]] Perform the following operations : a 2 · 1 0 2 1 0 − 1 b 4 · − 1 1 0 1 2 − 3 4 1 − 2 0 0 3. Multiplication E.6165 Definition: Let m , n , p be three non-zero natural numbers ( m;n;p ∈ N ∗ ) . Consider the two matrices A and B of dimensions m × n and n × p respectively. The product of the matrix A by the matrix B is a matrix of dimensions m × p whose coefficients c ij are defined by: c ij = n k =0 a ik × b kj ∀ i ∈ [[1 ; m ]] ; ∀ j ∈ [[1 ; p ]] Check the following equalities: a 1 2 0 1 3 1 · 1 2 = 5 2 5 b 3 1 1 1 2 1 · 1 1 2 1 = 5 4 3 2 4 3 c 1 2 4 1 · 3 1 1 2 = 5 5 13 6 E.5355 Perform the following matrix prod-ucts : a 2 1 − 1 3 · 4 − 1 0 2 b 2 − 1 1 0 − 2 1 1 0 1 · 2 − 1 1 1 3 − 1 0 1 1 E.3107 Perform the following matrix prod-ucts : a 2 1 − 1 − 3 1 0 · 0 4 1 − 1 2 1 b 5 2 − 1 2 · 2 − 1 − 3 1 E.5072 Perform the following matrix prod-ucts : a − 1 1 0 2 − 1 1 · 4 1 2 b − 1 2 1 0 2 2 − 2 3 1 · 1 − 2 1 https://chingmath.fr chapExoCorrec/6809 sacados/6809 chapExoCorrec/6155 sacados/6155 chapExoCorrec/6154 sacados/6154 chapExoCorrec/6165 sacados/6165 r641-0 chapExoCorrec/5355 sacados/5355 chapExoCorrec/3107 sacados/3107 chapExoCorrec/5072 sacados/5072
E.6157 Perform the following matrix prod-ucts : a 2 1 1 3 1 0 · 0 4 1 − 1 2 1 b 5 2 1 2 · 2 1 3 1 E.6158 Perform the following matrix prod-ucts : a − 1 1 0 2 1 1 · 4 1 2 b 1 2 1 0 2 2 2 3 1 · 1 2 1 E.6172 Perform the following matrix prod-ucts : a 1 2 0 1 · 1 0 3 2 1 1 b 1 1 0 1 2 2 · 1 1 − 1 1 E.6197 A company requires office supplies ev-ery year to run its ˇ administrative ı and ˇ productions ı depart-ments. The table below lists its requirements for one year in thousands of units : Rames de feuilles ( M 1 ) Stylo ( M 2 ) Tubes de colle ( M 3 ) Administration ( D 1 ) 3 4 1 Production ( D 2 ) 1 2 3 An invitation to tender is issued, to which two companies re-spond. The table below represents the unit price in euros of each of the supplies required by this company: Rames de feuilles ( M 1 ) Stylo ( M 2 ) Tubes de colle ( M 3 ) Fournisseur PasTropCher ( F 1 ) 2 3 1 Fournisseur BonPrix ( F 2 ) 3 1 2 1 a Write the matrix A = a ij où the coefficient a ij is the quantity of material M j required by the D i depart-ment. b Write the matrix B = b ij où the coefficient b ij is the price quoted by the supplier F j for the material M i . 2 Perform the following matrix product : A · B . 3 a In order to save money, which supplier should the company choose. b For this supplier, give the purchase amount for office supplies. 4. Multiplication and identity matrix E.6166 Definition: We call identity matrix of order n ( n ∈ N ∗ ) the square matrix of order n whose coefficients have the value 1 on the main diagonal and 0 elsewhere. It is noted I n . For I n = a ij , we have : a ii = 1 ∀ i ∈ [[1; n ]] a ij = 0 ∀ ( i;j ) ∈ [[1; n ]] 2 et i = j 1 Perform the following matrix products : a 1 2 3 4 · 1 0 0 1 b 1 0 0 0 1 0 0 0 1 · 1 2 3 4 5 6 7 8 9 2 What characteristic can be given to a matrix with only 1 on its main diagonal and zeros elsewhere? E.5144 We wish to determine the set of square matrices of dimensions 2 verifying: A 2 = I 2 To do this, consider the matrix A = a b c d où a , b , c and d are any four real numbers. 1 a Express the matrix A 2 in terms of the real numbers a , b , c and d . b Determine a system of equations verified by the reals a , b , c and d . 2 To study this problem, we’ll perform a case disjunction : a If b =0 , express the matrix A as a function of the real c . b If b =0 , express the matrix A as a function of the real a and b . 5. Commutativity of multiplication E.6167 Definition: Let A and B be two square matrices of order n ( n ∈ N ∗ ) , we say that the product of the matrices A and B is commu-tatif if : A · B = B · A https://chingmath.fr chapExoCorrec/6157 sacados/6157 chapExoCorrec/6158 sacados/6158 chapExoCorrec/6172 sacados/6172 chapExoCorrec/6197 sacados/6197 chapExoCorrec/6166 sacados/6166 chapExoCorrec/5144 sacados/5144 chapExoCorrec/6167 sacados/6167
1 Perform the following two matrix products : A 1 3 2 1 · 3 3 2 3 b 3 3 2 3 · 1 3 2 1 2 Perform the following two matrix products : a 2 1 1 2 · 0 3 2 3 b 0 3 2 3 · 2 1 1 2 3 Can we get a rule about the products A · B and B · A of two matrices? E.5076 Consider the following two matrices : A = 1 − 2 2 − 4 ; B = 2 6 1 3 1 Demonstrate that : A · B = 0 2 Determine the product matrix B · A . E.5093 For each question, show that the product of matrices A and B is commutative. a A = − 4 − 2 − 1 0 ; B = − 3 − 2 − 1 1 b A = − 4 − 2 2 4 ; B = − 2 − 1 1 2 E.8620 For each question, check whether the product of matrices A and B is commutative or not. a A = − 2 − 1 1 3 ; B = − 3 1 1 2 b A = 2 2 3 4 ; B = 1 2 3 3 6. Multiplication and solving systems of linear equations E.5073 Let a and b be two real numbers verifying the equality below : 5 2 − 1 3 · a b = 4 - 11 Determine the values of the reals a and b . E.8629 Let a and b be two real numbers verifying the equality below : − 1 3 2 1 · a b = − 2 11 Determine the values of the reals a and b . E.5074 Let a , b and c be three real numbers realizing the following equality: 3 1 4 2 − 1 0 1 − 1 1 · a b c = 5 0 − 1 Determine the values of the reals a , b c E.8630 Let a , b and c be three real numbers verifying the equality: 2 − 1 1 3 1 − 1 1 1 1 · a b c = 2 8 2 Determine the value of these reals. E.5075 Let a , b and c be solutions of equa-tion : 4 − 2 1 1 2 3 2 − 6 − 5 · a b c = 7 7 − 7 Determine the set of triplets ( a ; b ; c ) solutions of this equa-tion. E.8631 Let a , b and c be three real numbers that are solutions to the equation : − 1 3 1 1 0 2 − 1 6 4 · a b c = 5 7 6 Determine the set of triplets ( a ; b ; c ) solutions to this equa-tion. 7. Distributivity E.5078 Consider the following two matrices : A = 2 1 5 3 ; B = 1 6 − 1 2 1 Calculate: A + B . 2 a Calculate: A + B 2 b Calculate: A 2 +2 · A · B + B 2 3 Taking into account the algebraic properties of the ma- trix product, give the development of the expression A + B 2 . https://chingmath.fr chapExoCorrec/5076 sacados/5076 chapExoCorrec/5093 sacados/5093 chapExoCorrec/8620 sacados/8620 chapExoCorrec/5073 sacados/5073 chapExoCorrec/8629 sacados/8629 chapExoCorrec/5074 sacados/5074 chapExoCorrec/8630 sacados/8630 chapExoCorrec/5075 sacados/5075 chapExoCorrec/8631 sacados/8631 chapExoCorrec/5078 sacados/5078
E.5098 Let A be a square matrix of order n and the unit matrix I n of order n . 1 Expand the following expressions : a 2 · A + I n · I n − A b A + 2 · I n 2 2 Evaluate each of these two expressions in the case où : n =2 ; A = 4 − 1 2 − 2 E.5099 Let A and B be two matrices of order n whose product is commutative. 1 Expand the following expressions : a A + 2 · B · B − A b 2 · A − B 2 2 In the case of order 2 , evaluate the previous two expres-sions with the two matrices below whose product is com-mutative : A = 1 − 3 − 1 − 1 ; B = − 2 3 1 0 E.6824 We define the matrices A , B and M by: A = 0.8 0.8 0.2 0.2 ; B = 0.2 − 0.8 − 0.2 0.8 ; M = 0.9 0.4 0.1 0.6 1 Demonstrate that : M = A +0.5 · B 2 Check that A 2 = A , and that : A · B = B · A = 0 0 0 0 3 Demonstrate using reasoning by recurrence that, for any strictly positive natural number n : A n = A We admit that, for any strictly positive natural number n : B n = B . 4 Demonstrate that, for any natural number n : M n = A + 0.5 n · B . 8. Puissances n -ième of matrices E.8625 Consider the matrix: A = − 3 − 3 1 2 2 − 1 − 2 − 3 0 1 Determine the matrix A 2 . 2 Deduce the expression of A 101 E.8624 Consider the matrix: A = − 2 1 − 3 1 1 Determine the matrices A 2 and A 3 . 2 Deduce the expression of A 50 . E.3745 Consider the two matrices A and B defined by: A = 2 − 2 4 − 2 2 − 4 − 2 2 − 4 ; B = 1 − 3 − 4 − 1 3 4 1 − 3 − 4 1 Determine the square of each of these matrices. 2 For any natural number n , determine an expression for A n and an expression for B n . E.5100 Consider the matrix: A = 1 3 − 1 1 1 Establish equality: A 3 = − 8 · I 2 2 Deduce the expression of A 11 E.6069 Consider the two matrices A and B defined by: A = 1 1 1 1 1 1 1 1 1 ; B = k · A où k ∈ R . For what values of k the matrix B verifies the equality: B 2 = B 9. Puissances n -ième of matrices, cyclicity and congruence E.5129 Consider the matrix A defined by: A = 3 − 2 4 − 3 Determine the expression of the n -th power of the A matrix. E.5126 Consider the matrix A defined by: A = 1 − 4 − 4 − 1 1 2 1 − 2 − 3 Determine the expression of A n for any non-zero natural num-ber n . https://chingmath.fr chapExoCorrec/5098 sacados/5098 chapExoCorrec/5099 sacados/5099 chapExoCorrec/6824 sacados/6824 Extrait Liban Mai 2015 chapExoCorrec/8625 sacados/8625 chapExoCorrec/8624 sacados/8624 chapExoCorrec/3745 sacados/3745 chapExoCorrec/5100 sacados/5100 chapExoCorrec/6069 sacados/6069 chapExoCorrec/5129 sacados/5129 chapExoCorrec/5126 sacados/5126
E.8621 Consider the matrices A and B de-fined by: A = 1 − 3 − 3 − 2 2 3 2 − 3 − 4 ; B = − 2 − 1 1 1 0 − 1 − 2 − 2 1 1 Determine the squares of the matrices A and B . 2 For any natural integer n , determine an expression for A n and an expression for B n . E.8622 Consider the matrix A defined by: A = 1 − 1 3 − 2 Determine the expression of the n -th power of the A matrix. E.5127 Consider the matrix A defined by: A = 1 − 3 − 2 − 1 − 4 − 3 2 4 3 Determine the expression of A n for any non-zero natural num-ber n . E.8623 Consider the matrix A defined by: A = − 3 − 1 3 3 Determine the expression of the n -th power of the A matrix. E.5145 Consider the A square matrix of di-mension 3 defined by: A = 1 − 3 − 2 − 1 1 − 4 − 1 2 − 2 1 Establish that : A 3 =2 · I 3 . 2 Give an expression for A n for any natural number n . E.5128 Consider the matrix A defined by: A = 1 − 3 − 2 − 3 1 2 3 − 3 − 4 Determine the expression of A n for any non-zero natural num-ber n . 10. Puissances n -ième of matrices and reasoning by recurrence E.5082 Consider the matrix A = 1 1 0 1 Establish, that for any n ∈ N , we have : A n = 1 n 0 1 E.8626 Show that for any non-zero natural number n , we have : 4 − 4 3 − 3 n = 4 − 4 3 − 3 E.5115 Consider the sequence A defined by: A = 1 0 1 0 1 0 0 0 2 and the sequence u n defined by: u 0 = 0 ; u n +1 = u n + 2 n for all n ∈ N Establish that for any non-zero natural number n , we have equality: A n = 1 0 u n 0 1 0 0 0 2 n Note : use: A n +1 = A · A n E.7364 Consider the B square matrix of order 3 defined by: B = 5 12 1 4 1 3 5 12 1 4 1 3 1 6 1 2 1 3 Establish, using reasoning by recurrence, that for any integer n greater than or equal to 2 , the relation: B n = B 2 E.5083 Consider the matrix A = 2 0 1 1 Establish, that for any natural number n : A n = 2 n 0 2 n − 1 1 E.6100 Show that for any natural number n , we have : 5 − 4 3 − 2 n = 2 n +2 − 3 4 − 2 n +2 3 × 2 n − 3 4 − 3 × 2 n E.8627 Show that for any non-zero natural number n , we have : 2 4 1 2 n = 2 2 n − 1 2 2 n 2 2 n − 2 2 2 n − 1 E.8628 Show that for any non-zero natural number n , we have : 2 3 − 2 − 3 n = − 2 × ( − 1) n − 3 × ( − 1) n 2 × ( − 1) n 3 × ( − 1) n https://chingmath.fr chapExoCorrec/8621 sacados/8621 chapExoCorrec/8622 sacados/8622 chapExoCorrec/5127 sacados/5127 chapExoCorrec/8623 sacados/8623 chapExoCorrec/5145 sacados/5145 chapExoCorrec/5128 sacados/5128 chapExoCorrec/5082 sacados/5082 chapExoCorrec/8626 sacados/8626 chapExoCorrec/5115 sacados/5115 chapExoCorrec/7364 sacados/7364 chapExoCorrec/5083 sacados/5083 chapExoCorrec/6100 sacados/6100 chapExoCorrec/8627 sacados/8627 chapExoCorrec/8628 sacados/8628
E.6826 Consider the matrix: A = − 4 6 − 3 5 1 We call I the identity matrix of order 2 . Check that : A 2 = A +2 · I 2 Deduce an expression for A 3 and an expression for A 4 in the form ¸ · A + ˛ · I où ¸ and ˛ are real numbers. 3 Consider the sequences r n and s n defined by r 0 =0 and s 0 =1 and, for any natural number n : r n +1 = r n + s n s n +1 = 2 · r n Show that, for any natural number n : A n = r n · A + s n · I 11. Inverse matrix E.5094 Definition: Let n be a non-zero natural number and A and B two square matrices of order n . We say that the matrix B is the inverse matrix of the matrix A if : A · B = I n ; B · A = I n Consider the following two matrices A and B : A = − 1 1 3 1 − 2 − 5 − 1 2 4 ; B = − 2 − 2 − 1 − 1 1 2 0 − 1 − 1 1 Determine the matrices of the products A · B and B · A . 2 What can be said about the matrices A and B ? E.5095 Consider the following two matrices A and B : A = 1 − 2 − 2 − 2 1 2 − 1 1 1 ; B = − 1 0 − 2 0 − 1 2 − 1 1 − 3 1 Determine products A · B and B · A . 2 What can be said about the matrices A and B ? E.5096 Consider the following two matrices A and B : A = 1 − 2 − 2 − 2 1 0 − 2 2 2 ; B = − 2 0 − 2 − 4 2 − 4 2 − 2 3 1 Determine products A · B and B · A . 2 Deduce the inverse matrix of the matrix A . 12. Inverse matrix and algebraic expressions E.5116 Consider the square matrix A of or-der 3 defined by: A = 1 − 2 − 2 − 1 0 − 2 − 1 − 2 0 1 a Determine the matrix A 2 + A . b Express the inverse matrix of A as a function of A and I 3 . 2 Deduce the inverse matrix of matrix A . E.5117 Consider the square matrix A of or-der 3 defined by: A = 1 − 2 − 1 − 2 1 1 − 2 2 0 1 a Determine the matrix A 2 − 3 × A . b Express the inverse matrix of A as a function of A and I 3 . 2 Deduce the inverse matrix of matrix A . E.6871 Consider the square matrix A of or-der 3 defined by: A = 1 − 4 − 2 − 2 3 − 1 − 4 − 4 3 1 a Determine the matrix A 2 . b Determine matrix A 2 − 2 · A . c Deduce an expression for the inverse matrix of the A matrix as a function of A and I 3 . 2 Deduce the inverse matrix of matrix A . https://chingmath.fr chapExoCorrec/6826 sacados/6826 chapExoCorrec/5094 sacados/5094 chapExoCorrec/5095 sacados/5095 chapExoCorrec/5096 sacados/5096 chapExoCorrec/5116 sacados/5116 chapExoCorrec/5117 sacados/5117 chapExoCorrec/6871 sacados/6871
E.6825 We give the matrices : M = 1 1 1 1 − 1 1 4 2 1 ; I = 1 0 0 0 1 0 0 0 1 1 Determine the matrix M 2 . We give : M 3 = 20 10 11 12 2 9 42 20 21 2 Verify that : M 3 = M 2 +8 · M +6 · I 3 3 Deduce that M is invertible and that : M − 1 = 1 6 · M 2 − M − 8 · I 3 4 Deduce the inverse matrix of matrix M . 13. Inverse matrices of order 2 E.8634 Definition: Consider A the matrix defined by: A = a b c d We call determinant of the matrix A , denoted det( A ) , the number: det( A ) = a · d − b · d Determine the determinant of the following matrices : A = 1 3 3 − 1 ; B = 2 4 − 3 2 ; C = 6 3 4 2 E.5097 Proposition: Consider the matrix A = a b c d . The matrix A is invert-ible if, and only if, det( A ) =0 . Then : A − 1 = 1 det( A ) · d − b − c a Determine the inverses of the following matrices : A − 3 2 − 2 2 b − 2 − 1 − 1 − 1 c 3 4 4 4 E.6828 Establish that each of the matri-ces below are invertible and determine the expression of their inverse matrices : a 1 − 1 − 3 2 b 2 1 − 1 − 1 c 3 − 1 − 1 0 E.8633 Determine the inverses of the follow-ing matrices : A 1 2 1 3 b 1 0 − 4 − 4 c 2 0 3 − 4 E.5112 1 Let n be a non-zero natural number. Consider A and B two square matrices of order n invertible and – a non-zero real number. a Show that the product A · B is an invertible matrix whose inverse is to be specified. b Show that the matrix – · A is an invertible matrix whose inverse will be specified. 2 Consider the following two matrices A and B : A = 4 3 1 0 ; B = 3 − 4 − 2 3 Show that the following relationship is false : A + B − 1 = A − 1 + B − 1 https://chingmath.fr chapExoCorrec/6825 sacados/6825 chapExoCorrec/8634 sacados/8634 chapExoCorrec/5097 sacados/5097 chapExoCorrec/6828 sacados/6828 chapExoCorrec/8633 sacados/8633 chapExoCorrec/5112 sacados/5112