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Lista de problemas ÁLGEBRA LINEAL
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Lista de problemas de ÁLGEBRA LINEAL de 2020/2021 con exámenes de años anteriores.
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Col·lecció de problemes d’àlgebra linealEnginyeria quı́mica i enginyeria de materials
Universitat de Barcelona
Curs 2020–2021
1
,Problemes d’Àlgebra Lineal Grau d’Enginyeria Quı́mica
Curs 2020–2021 Grau d’Enginyeria de Materials
Continguts
1 Matrius i sistemes d’equacions 3
2 Determinants 6
3 Espais Vectorials 9
4 Aplicacions Lineals 13
5 Nombres Complexos 17
6 Polinomis 18
7 Diagonalització 19
8 Producte escalar euclidi 22
9 Exàmens i controls 24
,Problemes d’Àlgebra Lineal Grau d’Enginyeria Quı́mica
Curs 2020–2021 Grau d’Enginyeria de Materials
1 Matrius i sistemes d’equacions
1. Donades les matrius següents, indiqueu quantes files i quantes columnes té cadascuna i decidiu
quines parelles es poden sumar i quines es poden multiplicar, i en quin ordre. Efectueu totes
les operacions possibles.
1 1 7
A = 2 , B = 5 2 −1 , C = −2 4 ,
3 0 3
7 5 2 −1
1 7 2
D = 0 , E = 3 3 3 , F = .
−2 4 1
0 0 0 0
2. Amb les matrius del problema anterior, decidiu si les operacions següents són possibles i, en
cas afirmatiu, efectueu-les.
A · DT + E + F · C, (A + D) · E, B · E · C · F, (B + E) · C.
3. Demostreu que el producte de matrius diagonals és commutatiu. Doneu un exemple de dues
matrius triangulars que no commutin entre elles.
4. Calculeu el rang de les matrius següents utilitzant el mètode de Gauss. Decidiu si tenen inversa
i, en cas afirmatiu, calculeu-la.
1 1 −1 1
2 3 4 −2 1 −1 1 1 1
1 −1 −2 1 −1 ,
5 , 0 1 0 , 0 1 1 0
4 1 0 8 1 1 −1
1 0 0 1
1 2 1 1 −1 2 −2 1 1
−2 1 −2 , −1 2 1 , 0 1 −1 .
1 0 1 −1 3 4 1 1 1
5. Decidiu si les matrius següents tenen inversa i, en cas afirmatiu, calculeu-la.
2 0 5 0 2 3 0 1 1 1 2 1
, , , , , .
0 3/2 0 0 0 1 0 1 −1 2 4 2
6. Donats els sistemes d’equacions següents, escriviu-los en forma matricial, decidiu si tenen
solució i, en cas afirmatiu, trobeu totes les solucions.
x + y + z = 2 x + y + z = 2 x − y + z = 2
2x + 3y − 2z = 7 , 2x + 3y − 2z = 7 , 2y − 3z = 7 ,
−x − 5z = 3 −x − 5z = 1 −x + y − 2z = 3
x + y + 2z − t = 1
2x − y + 2z − t = 2
x + y + 2z + t = −1
, x + y − 2z + 2t = 3 .
x + y − z + 3t = 0
−x + 2y − 4z + 3t = 1
−x + 2y − 2z = −2
, Problemes d’Àlgebra Lineal Grau d’Enginyeria Quı́mica
Curs 2020–2021 Grau d’Enginyeria de Materials
7. Trobeu els valors del paràmetre m que fan compatible el sistema d’equacions
5x + 3y = 3
5x + 2y = 2 .
5x + my = 2
8. Discutiu, segons els valors dels paràmetres a i b, els sistemes d’equacions següents:
x + y + az = 1
ax + y + az = 1
(i) x − ay + z = −1 (ii) x−y+z =b
x + ay + z = b ax + (a − 1)y − z = −1
9. Discutiu i resoleu els sistemes d’equacions lineals següents:
x − 3y + 2z = 6
2x − y + 3z + t − 3u = 2
(i) 2x + y − 5z = −4 3x − 2y + z − t − 2u = 4
(iii)
2x − 13y + 13z = 28
4x + 5y − z − 3t − u = 6
x − 2y + 2z + 2t + 4u = −3
x + y + iz + t = 0
2x − y + 2z − t = 1
3x + 2y + z = 1
(iv)
(ii) 5x + 3y + 4z = 2
x + iy − z + it = 2
x+y−z =1 x+y+z−t=0
8x + 6y + 5z + 2t = 21
x + 2y + z + t = 3
3x + 3y + 2z + t = 10
x + 4y + 5z + 2t = 2
(v) 4x + 2y + 3z + t = 8 (vii) +2y + 4z + t = −1
7x + 4y + 5z + 2t = 18
x − 3z = 4
3x + 5y + z + t = 15 4x + 6y + 3t = 13
2x + 3y + z + 2t = 4
x + 2y + z + t = 1
4x + 3y + z + t = 5
x + 4y + 5z + 2t = 1
(vi) 5x + 11y + 3z + 2t = 2 (viii) +2y + 4z + t = 1
2x + 5y + z + t = 1
x − 3z = 1
x − 7y − z + 2t = 7 4x + 6y + 3t = 1
10. Discutiu i resoleu els sistemes següents, segons els valors dels paràmetres:
2x + y − z = 3
x + 3y = 2a
(i) x + my + z = 3 (iii) x+y =5
3x + y − mz = 4 2ax + 6y = a + 3
2x − ay = 1
x + my + z = 1
(ii) −x + 2y − az = 1 (iv) mx + y + (m − 1)z = m
−y + 2z = 1 x+y+z =m+1
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