Homework-1 /Linear Algebra.
/2024-2025 academic year/ by Jandigulov AR
Content
1. Solving a system of linear equations .......................................................................................................... 2
2. Linear independence.................................................................................................................................... 4
3. A matrix equation ........................................................................................................................................ 5
1. Solving a system of linear equations
Solve a system of linear equations in three ways:
a) by the RREF method;
b) according to Kramer's formulas;
c) using the inverse matrix
4 − 3 2 10 − 2 1 2 3
1. 3 − 1 −2 3 2. − 3 6 − 2 − 12
5 2 25 −3 6
3 4 2
1 − 2 5 6 5 1 1 21
3. 3 2 5 38 4. − 1 1 2 6
3 2 32 −1 5 3 21
2
2 − 2 − 2 − 10 5 1 − 1 22
5. − 3 4 1 13 6. − 3 5 5 20
1 − 5 − 3 − 37 5 − 3 5 44
1 2 6 36 6 − 3 − 5 − 12
7. 3 2 6 44 8. 1 4 1 36
3 −1 − 2 0 − 2 2 6
1
2 2 3 22 6 2 − 1 27
9. − 5 − 3 4 − 26 10. 4 − 2 − 3 1
− 5 5 −1 − 12 6 − 5 − 5 − 6
5 3 2 60 3 − 5 − 3 − 12
11. − 3 − 1 5 6 12. − 5 1 3 −8
1 42 6 42
5 1 5 1
1 −5 −2 − 22 1 − 3 3 5
13. − 5 4 3 12 14. 3 − 2 −1 − 7
5 −3 2 5 2 −5 − 7
2
3 −2 6 30 1 1 −3 3
15. 1 1 3 18 16. 3 − 1 6 19
3 −3 12 −1 1 3
1 1
2
− 2 1 2 0 1 5 − 3 15
17. 2 4 3 20 18. 4 5 2 54
3 − 3 5 13 1 1 − 5 − 21
4 − 1 2 27 1 −2 − 1 − 10
19. 5 − 1 2 33 20. 3 2 − 1 10
2 3 2 27 − 5 2 2
1
6 − 5 5 40 5 −3 1 21
21. 1 − 3 − 5 − 21 22. 2 3 2 29
3 6 − 2 19 −1 2 8
1
−1 − 5 5 4 2 −1 −1 4
23. 5 3 5 56 24. 5 −3 − 2 10
1 1 4 24 1 −1 − 5 − 13
3
2. Linear independence
Will vector 𝐛 be a linear combination of vectors {a1 , a2 , a2 }?
4
3. A matrix equation
Solve the matrix equation .
Check the answer by substituting it into the equation.
5
6
