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ICTE124: INTRODUCTION TO ALGEBRA PAST QUESTIONS

August 28, 2022
2 years ago
Adaora Okonkwo Avatar
ICTE124: INTRODUCTION TO ALGEBRA PAST QUESTIONS

  

Good day to each and every one of you who is present.

Let's honor our knowledge by responding to a few straightforward questions since "information becomes power only when we put it into some use."

The adage "Knowledge has a beginning but no end" is true, so let's keep learning.

I appreciate it.

1. If a matrix is in reduced row echelon form, then it is also in row echelon form.

TRUE / FALSE

2. If an elementary row operation is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form.

TRUE / FALSE

3. Every matrix has a unique row echelon form.

TRUE / FALSE

4. A homogeneous linear system in n unknowns whose corresponding augmented matrix has a reduced row echelon form with r leading 1's has n−r free variables.

TRUE / FALSE


5. All leading 1's in a matrix in row echelon form must occur in different columns.

TRUE / FALSE

6. If every column of a matrix in row echelon form has a leading 1, then all entries that are not leading 1's are zero.

TRUE / FALSE
7. If a homogeneous linear system of n equations in n unknowns has a corresponding augmented matrix with a reduced row echelon form containing n leading 1's, then the linear system has only the trivial solution.

TRUE / FALSE

8. If the reduced row echelon form of the augmented matrix for a linear system has a row of zeros, then the system must have infinitely many solutions.

TRUE / FALSE

9. If a linear system has more unknowns than equations, then it must have infinitely many solutions.

TRUE / FALSE




10. The determinant of the 2 × 2 matrix a b c d is ad + bc.

TRUE / FALSE

11. Two square matrices that have the same determinant must have the same size.

TRUE / FALSE

12. The minor Mij is the same as the cofactor Cij if i + j is even.

TRUE / FALSE

13. If A is a 3 × 3 symmetric matrix, then Cij = Cji for all i and j.

TRUE / FALSE

14. The number obtained by a cofactor expansion of a matrix A is independent of the row or column chosen for the expansion.

TRUE / FALSE

15. If A is a square matrix whose minors are all zero, then det(A) = 0.

TRUE / FALSE

16. The determinant of a lower triangular matrix is the sum of the entries along the main diagonal.

TRUE / FALSE


17. For every square matrix A and every scalar c, it is true that det(cA) = c det(A).

TRUE / FALSE

18. For all square matrices A and B, it is true that det(A + B) = det(A) + det(B).

TRUE / FALSE

19. For every 2 × 2 matrix A it is true that det(A2 ) = (det(A)) 2.
TRUE / FALSE

20. If A is a 4 × 4 matrix and B is obtained from A by interchanging the first two rows and then interchanging the last two rows, then det(B) = det(A).
TRUE / FALSE

21. If A is a 3 × 3 matrix and B is obtained from A by multiplying the first column by 4 and multiplying the third column by 3 4 , then det(B) = 3 det(A).
TRUE / FALSE

22. If A is a 3 × 3 matrix and B is obtained from A by adding 5 times the first row to each of the second and third rows, then det(B) = 25 det(A).
TRUE / FALSE

23. If A is an n × n matrix and B is obtained from A by multiplying each row of A by its row number, then det(B) = n(n + 1) 2 det(A)
TRUE / FALSE

24. If A is a square matrix with two identical columns, then det(A) = 0.
TRUE / FALSE

25. If the sum of the second and fourth row vectors of a 6 × 6 matrix A is equal to the last row vector, then det(A) = 0.
TRUE / FALSE

26. If A is a 3 × 3 matrix, then det(2A) = 2 det(A).
TRUE / FALSE

27. If A and B are square matrices of the same size such that det(A) = det(B), then det(A + B) = 2 det(A).
TRUE / FALSE

28. If A and B are square matrices of the same size and A is invertible, then det(A −1 BA) = det(B).
TRUE / FALSE

29. A square matrix A is invertible if and only if det(A) = 0.
TRUE / FALSE

30. The matrix of cofactors of A is precisely [adj(A)] T.
TRUE / FALSE

31. For every n × n matrix A, we have A · adj(A) = (det(A))I n.
TRUE / FALSE

32. If A is a square matrix and the linear system Ax = 0 has multiple solutions for x, then det(A) = 0.
TRUE / FALSE

33. If A is an n × n matrix and there exists an n × 1 matrix b such that the linear system Ax = b has no solutions, then the reduced row echelon form of A cannot be I n.
TRUE / FALSE

34. If E is an elementary matrix, then Ex = 0 has only the trivial solution.
TRUE / FALSE

35. If A is an invertible matrix, then the linear system Ax = 0 has only the trivial solution if and only if the linear system A −1 x = 0 has only the trivial solution.
TRUE / FALSE

36. If A is invertible, then adj(A) must also be invertible.
TRUE / FALSE

37. If A has a row of zeros, then so does adj(A).
TRUE / FALSE

38. A set containing a single vector is linearly independent.
TRUE / FALSE

39. The set of vectors {v, kv} is linearly dependent for every scalar k.
TRUE / FALSE

40. Every linearly dependent set contains the zero vector.
TRUE / FALSE

41. If the set of vectors {v 1, v 2, v 3} is linearly independent, then {kv 1, kv 2, kv 3} is also linearly independent for every nonzero scalar k.
TRUE / FALSE

42. If v 1,..., v n are linearly dependent nonzero vectors, then at least one vector v k is a unique linear combination of v 1,..., v k−1.
TRUE / FALSE

43. The set of 2 × 2 matrices that contain exactly two 1's and two 0's is a linearly independent set in M22.
TRUE / FALSE

44. The three polynomials (x−1)(x + 2), x(x + 2), and x(x−1) are linearly independent.
TRUE / FALSE

45. The functions f1 and f2 are linearly dependent if there is a real number x such that k 1 f1(x) + k 2 f2(x) = 0 for some scalars k 1 and k 2.
TRUE / FALSE








Hope this is helpful. I'll be the smart one, please.