Đề thi, bài tập trắc nghiệm online Đại số tuyến tính – Đề 3

1. For a square matrix A, if λ is an eigenvalue, what is always true about A - λI?

A. A - λI is invertible.

B. A - λI is the zero matrix.

C. The determinant of (A - λI) is zero.

D. The rank of (A - λI) is equal to the dimension of A.

2. What does the Gram-Schmidt process achieve?

A. Diagonalizes a matrix.

B. Finds eigenvalues and eigenvectors.

C. Transforms a basis into an orthonormal basis.

D. Solves systems of linear equations.

3. Which of the following is a property of eigenvalues of a real symmetric matrix?

A. Eigenvalues must be complex numbers.

B. Eigenvalues must be positive integers.

C. Eigenvalues must be real numbers.

D. Eigenvalues must be zero.

4. Which of the following is NOT a subspace of R²?

A. The set of all vectors of the form (x, 0).

B. The set of all vectors of the form (x, x).

C. The set of all vectors of the form (x, 1).

D. The set of all vectors of the form (0, y).

5. Consider a linear transformation T: R² → R². If T rotates every vector by 90 degrees counterclockwise, what is the matrix representation of T?

A. [[0, 1], [1, 0]]

B. [[0, -1], [1, 0]]

C. [[1, 0], [0, 1]]

D. [[-1, 0], [0, -1]]

6. If A is an n x n matrix and c is a scalar, how does det(cA) relate to det(A)?

A. det(cA) = c * det(A)

B. det(cA) = cⁿ * det(A)

C. det(cA) = det(A) + c

D. det(cA) = det(A)ᶜ

7. What is a linear transformation?

A. A function that maps vectors to scalars.

B. A function between two vector spaces that preserves vector addition and scalar multiplication.

C. Any function between vector spaces.

D. A function that maps scalars to vectors.

8. What is the dimension of the vector space of all m x n matrices?

A. m + n

B. m - n

C. m * n

D. max(m, n)

9. If A and B are n x n matrices, which of the following is generally NOT true?

A. det(AB) = det(A)det(B)

B. det(A + B) = det(A) + det(B)

C. det(Aᵀ) = det(A)

D. det(kA) = kⁿdet(A) where k is a scalar

10. For a system of linear equations Ax = b to have at least one solution, what must be true about the ranks of the augmented matrix [A|b] and the coefficient matrix A?

A. rank(A) must be less than rank([A|b]).

B. rank(A) must be greater than rank([A|b]).

C. rank(A) must be equal to rank([A|b]).

D. There is no condition on the ranks for a solution to exist.

11. If a square matrix A is invertible, what is the inverse of its transpose, (Aᵀ)⁻¹?

A. (A⁻¹)ᵀ

B. -(A⁻¹)ᵀ

C. Aᵀ

D. -Aᵀ

12. Which of the following is a condition for a square matrix to be invertible?

A. Its determinant is zero.

B. Its rank is less than its dimension.

C. Its null space contains only the zero vector.

D. It has linearly dependent columns.

13. Which of the following is NOT a property of vector spaces?

A. Closure under vector addition.

B. Closure under scalar multiplication.

C. Commutativity of scalar multiplication (c(dv) = (cd)v).

D. Closure under matrix multiplication.

14. Which of the following sets of vectors in R³ is linearly independent?

A. {(1, 0, 0), (2, 0, 0), (0, 1, 0)}

B. {(1, 1, 0), (0, 1, 1), (1, 2, 1)}

C. {(1, 0, 1), (0, 1, 0), (0, 0, 1), (1, 1, 1)}

D. {(1, 2, 3), (0, 1, 2), (0, 0, 1)}

15. What is the dot product of two vectors also known as?

A. Cross product.

B. Scalar product.

C. Vector product.

D. Matrix product.

16. Given two vectors u = (1, 2) and v = (3, k). For what value of k are vectors u and v orthogonal?

A. 6

B. -6

C. 3/2

D. -3/2

17. Given a matrix A, under what condition is A = A⁻¹?

A. When A is the identity matrix.

B. When A is the zero matrix.

C. When A² = I (where I is the identity matrix).

D. When A is a diagonal matrix.

18. If matrix A is a square matrix and its determinant is zero, which of the following statements is always true?

A. The matrix A is invertible.

B. The matrix A has full rank.

C. The system of linear equations Ax = 0 has only the trivial solution.

D. The system of linear equations Ax = 0 has infinitely many solutions.

19. Which of the following is true about the column space of a matrix A?

A. It is always equal to the row space of A.

B. It is a subspace of the column space of Aᵀ.

C. It is the set of all possible linear combinations of the columns of A.

D. It is always orthogonal to the null space of A.

20. Which of the following operations is NOT always valid for matrix multiplication?

A. Associativity: (AB)C = A(BC)

B. Distributivity over addition: A(B + C) = AB + AC

C. Commutativity: AB = BA

D. Scalar multiplication: c(AB) = (cA)B = A(cB)

21. What is the determinant of a matrix if two of its rows are identical?

A. Equal to 1.

B. Equal to -1.

C. Equal to 0.

D. Equal to the product of the diagonal elements.

22. What is the rank of a matrix?

A. The number of rows in the matrix.

B. The number of columns in the matrix.

C. The dimension of the null space of the matrix.

D. The dimension of the column space (or row space) of the matrix.

23. Which of the following describes an eigenvector of a matrix A?

A. A vector that is transformed into the zero vector when multiplied by A.

B. A vector that, when multiplied by A, only changes in magnitude, not direction.

C. A vector that is orthogonal to all columns of A.

D. A vector that is a linear combination of the rows of A.

24. Which of the following is NOT a basic operation when solving a system of linear equations using Gaussian elimination?

A. Swapping two rows.

B. Multiplying a row by a non-zero scalar.

C. Adding a multiple of one row to another row.

D. Multiplying two rows together.

25. What is the characteristic polynomial of a square matrix A used to find?

A. Eigenvectors.

B. Eigenvalues.

C. Determinant.

D. Trace.

26. What is the cofactor of an element aᵢⱼ in a matrix A?

A. The determinant of the matrix obtained by deleting the i-th row and j-th column.

B. The negative of the determinant of the matrix obtained by deleting the i-th row and j-th column.

C. (-1)ⁱ⁺ʲ times the determinant of the matrix obtained by deleting the i-th row and j-th column.

D. The element aⱼᵢ.

27. What is the determinant of an elementary matrix resulting from swapping two rows of the identity matrix?

A. 1

B. -1

C. 0

D. 2

28. What is the null space of a matrix A?

A. The set of all vectors b such that Ax = b has a solution.

B. The set of all vectors x such that Ax = 0.

C. The set of all linear combinations of the columns of A.

D. The set of all linear combinations of the rows of A.

29. If matrix A is diagonalizable, it means that A is similar to which type of matrix?

A. An upper triangular matrix.

B. A lower triangular matrix.

C. A diagonal matrix.

D. An identity matrix.

30. What is the trace of a square matrix?

A. The determinant of the matrix.

B. The sum of all elements in the matrix.

C. The sum of the diagonal elements of the matrix.

D. The product of the diagonal elements of the matrix.