Test of consistency and solution of linear systems using matrices - Question Bank
1. If a system of linear equations Ax = b is consistent, what is the relationship between rank(A) and rank([A|b])?
2. The condition for a system of n linear equations in n variables to have a non-trivial solution is:
3. For a system of n equations in n unknowns, if the rank of the coefficient matrix is n-1 and the rank of the augmented matrix is also n-1, the system has:
4. If det(A) = 0 and the system Ax = b is inconsistent, then rank(A) must be:
5. The system x = 0, y = 0 is an example of a:
6. If rank(A) = r, rank([A|b]) = r, and r < n (number of variables), the system Ax = b has:
7. For the system x + y + z = 1, x - y + z = 1, x + y - z = 1. The determinant of the coefficient matrix is:
8. If a system of linear equations has infinitely many solutions, it is:
9. A system of linear equations Ax = b is consistent if rank(A) is:
10. Consider the system: x + y + z = 1, 2x + 2y + 2z = 2, 3x + 3y + 3z = 3. The rank of the augmented matrix [A|b] is:
11. If rank(A) = n (number of variables) and rank([A|b]) = n, the system Ax = b has:
12. Which of the following systems represents a homogeneous system of linear equations?
13. The number of non-trivial solutions for a homogeneous system Ax = 0 with n variables and rank(A) = r is:
14. If a system of linear equations has no solution, it is called:
15. For a system of n linear equations in n variables, if det(A) = 0, the system is guaranteed to have:
16. Consider the system: x + y + z = 6, x + 2y + 3z = 10, 3x + 5y + 7z = 26. If rank(A) = 2 and rank([A|b]) = 3, the system has:
17. Cramer's rule can be used to solve a system of linear equations Ax = b if:
18. If rank(A) = r, rank([A|b]) = r, and r < number of variables, the system Ax = b has:
19. Which of the following represents the augmented matrix for the system: x + y = 5, 2x - z = 1?
20. The system x + y + z = 0, 2x + 2y + 2z = 0, 3x + 3y + 3z = 0 has:
21. If rank(A) = rank([A|b]) = r, and r is the number of variables, the system Ax = b is:
22. For a homogeneous system Ax = 0 where A is an n x n matrix, if rank(A) = n, then the system has:
23. A system of linear equations is inconsistent if and only if:
24. Consider the system: x + 2y = 3, 2x + 4y = 5. The rank of the coefficient matrix A is:
25. If the system Ax = b is consistent and has a unique solution, then rank(A) must be equal to:
26. For a system of m linear equations in n variables, if rank(A) = m and rank([A|b]) = m, and m < n, the system has:
27. The system x = 1, y = 2, x + y = 3 can be represented in matrix form as:
28. If a system Ax = b has infinitely many solutions, then rank(A) is:
29. Consider the system: x + y = 2, x + y = 3. This system is:
30. For a system of 3 linear equations in 3 variables, if det(A) = 0 and the system is consistent, how many free variables are there?
31. Which method is commonly used to determine the rank of a matrix for consistency checks?
32. If rank(A) = 1, rank([A|b]) = 1, and there are 3 variables, the system Ax = b has:
33. In the context of linear systems, what does it mean for a system to be homogeneous?
34. The system of equations x + y + z = 6, x + 2y + 3z = 14, 2x + 3y + 4z = 20 is:
35. If rank(A) = rank([A|b]) = r and r = n (number of variables), the system Ax = b has:
36. For the system x - y = 1, 2x - 2y = 2, the rank of the coefficient matrix A is:
37. If a system of linear equations Ax = b has a unique solution, then rank(A) must be:
38. Which condition guarantees a unique solution for a system of n linear equations in n variables Ax = b?
39. For a system of n linear equations in n variables, represented by Ax = b, if det(A) = 0 and the system is consistent, then it has:
40. Consider the system: x + y + z = 1, x + 2y + 3z = 4, 3x + 5y + 7z = 10. If rank(A) = 2 and rank([A|b]) = 3, the system has:
41. The number of free variables in a consistent system of n linear equations in m variables, where rank(A) = r, is given by:
42. If det(A) is non-zero for a homogeneous system Ax = 0, then the system has:
43. For the system of equations: x + y = 3, 2x + 2y = 6. The determinant of the coefficient matrix is:
44. A system of linear equations is said to be consistent if it has:
45. Which matrix represents the augmented matrix for the system of equations 2x - y + 3z = 9, x + y - z = 2, 3x - y + 2z = 7?
46. If rank(A) = rank([A|b]) = r, and r < n (where n is the number of variables), then the system Ax = b has:
47. For a homogeneous system of linear equations Ax = 0, where A is a square matrix, a non-trivial solution exists if and only if:
48. If the rank of matrix A is r, and the rank of the augmented matrix [A|b] is r+1, then the system of linear equations Ax = b has:
49. Consider the system of linear equations: x + 2y = 5 and 2x + 4y = 10. The system has:
50. For a system of linear equations Ax = b, if the rank of the coefficient matrix A is equal to the rank of the augmented matrix [A|b] and this rank is equal to the number of variables, then the system has: