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.
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