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

A A. A - λI is invertible. B B. A - λI is the zero matrix. C C. The determinant of (A - λI) is zero. D D. The rank of (A - λI) is equal to the dimension of A.

A A. Diagonalizes a matrix. B B. Finds eigenvalues and eigenvectors. C C. Transforms a basis into an orthonormal basis. D D. Solves systems of linear equations.

A A. Eigenvalues must be complex numbers. B B. Eigenvalues must be positive integers. C C. Eigenvalues must be real numbers. D D. Eigenvalues must be zero.

A A. The set of all vectors of the form (x, 0). B B. The set of all vectors of the form (x, x). C C. The set of all vectors of the form (x, 1). D D. The set of all vectors of the form (0, y).

A A. [[0, 1], [1, 0]] B B. [[0, -1], [1, 0]] C C. [[1, 0], [0, 1]] D D. [[-1, 0], [0, -1]]

A A. det(cA) = c * det(A) B B. det(cA) = cⁿ * det(A) C C. det(cA) = det(A) + c D D. det(cA) = det(A)ᶜ

A A. A function that maps vectors to scalars. B B. A function between two vector spaces that preserves vector addition and scalar multiplication. C C. Any function between vector spaces. D D. A function that maps scalars to vectors.

A A. m + n B B. m - n C C. m * n D D. max(m, n)

A A. det(AB) = det(A)det(B) B B. det(A + B) = det(A) + det(B) C C. det(Aᵀ) = det(A) D D. det(kA) = kⁿdet(A) where k is a scalar

A A. rank(A) must be less than rank([A|b]). B B. rank(A) must be greater than rank([A|b]). C C. rank(A) must be equal to rank([A|b]). D D. There is no condition on the ranks for a solution to exist.

A A. (A⁻¹)ᵀ B B. -(A⁻¹)ᵀ C C. Aᵀ D D. -Aᵀ

A A. Its determinant is zero. B B. Its rank is less than its dimension. C C. Its null space contains only the zero vector. D D. It has linearly dependent columns.

A A. Closure under vector addition. B B. Closure under scalar multiplication. C C. Commutativity of scalar multiplication (c(dv) = (cd)v). D D. Closure under matrix multiplication.

A A. {(1, 0, 0), (2, 0, 0), (0, 1, 0)} B B. {(1, 1, 0), (0, 1, 1), (1, 2, 1)} C C. {(1, 0, 1), (0, 1, 0), (0, 0, 1), (1, 1, 1)} D D. {(1, 2, 3), (0, 1, 2), (0, 0, 1)}

A A. Cross product. B B. Scalar product. C C. Vector product. D D. Matrix product.

A A. 6 B B. -6 C C. 3/2 D D. -3/2

A A. When A is the identity matrix. B B. When A is the zero matrix. C C. When A² = I (where I is the identity matrix). D D. When A is a diagonal matrix.

A A. The matrix A is invertible. B B. The matrix A has full rank. C C. The system of linear equations Ax = 0 has only the trivial solution. D D. The system of linear equations Ax = 0 has infinitely many solutions.

A A. It is always equal to the row space of A. B B. It is a subspace of the column space of Aᵀ. C C. It is the set of all possible linear combinations of the columns of A. D D. It is always orthogonal to the null space of A.

A A. Associativity: (AB)C = A(BC) B B. Distributivity over addition: A(B + C) = AB + AC C C. Commutativity: AB = BA D D. Scalar multiplication: c(AB) = (cA)B = A(cB)

A A. Equal to 1. B B. Equal to -1. C C. Equal to 0. D D. Equal to the product of the diagonal elements.

A A. The number of rows in the matrix. B B. The number of columns in the matrix. C C. The dimension of the null space of the matrix. D D. The dimension of the column space (or row space) of the matrix.

A A. A vector that is transformed into the zero vector when multiplied by A. B B. A vector that, when multiplied by A, only changes in magnitude, not direction. C C. A vector that is orthogonal to all columns of A. D D. A vector that is a linear combination of the rows of A.

A A. Swapping two rows. B B. Multiplying a row by a non-zero scalar. C C. Adding a multiple of one row to another row. D D. Multiplying two rows together.

A A. Eigenvectors. B B. Eigenvalues. C C. Determinant. D D. Trace.

A A. The determinant of the matrix obtained by deleting the i-th row and j-th column. B B. The negative of the determinant of the matrix obtained by deleting the i-th row and j-th column. C C. (-1)ⁱ⁺ʲ times the determinant of the matrix obtained by deleting the i-th row and j-th column. D D. The element aⱼᵢ.

A A. 1 B B. -1 C C. 0 D D. 2

A A. The set of all vectors b such that Ax = b has a solution. B B. The set of all vectors x such that Ax = 0. C C. The set of all linear combinations of the columns of A. D D. The set of all linear combinations of the rows of A.

A A. An upper triangular matrix. B B. A lower triangular matrix. C C. A diagonal matrix. D D. An identity matrix.

A A. The determinant of the matrix. B B. The sum of all elements in the matrix. C C. The sum of the diagonal elements of the matrix. D D. The product of the diagonal elements of the matrix.