Virtual Labs

1. Let A be a real n×n matrix. If A is both orthogonal and symmetric, what must A be?

a: A diagonal matrix with entries ±1 Explanation

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b: A rotation matrix Explanation

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c: A nilpotent matrix Explanation

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d: A skew-symmetric matrix Explanation

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2. If A is a non-zero nilpotent matrix, what is true about its determinant and trace?

a: Both are non-zero Explanation

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b: Trace is zero, determinant is zero Explanation

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c: Trace is non-zero, determinant is zero Explanation

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d: Trace is zero, determinant is non-zero Explanation

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3. Let A be a square matrix such that A is similar to a diagonal matrix D. What can be said about the minimal polynomial of A?

a: It must be (x - λ₁)...(x - λₙ) Explanation

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b: It must be the same as the characteristic polynomial Explanation

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c: It must split into linear factors over ℂ Explanation

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d: It must be irreducible Explanation

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4. Let A be an n×n matrix over ℂ such that Aⁿ = I. Which of the following must be true about the eigenvalues of A?

a: They are all real Explanation

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b: They are nth roots of unity Explanation

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c: They are all equal to 1 Explanation

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d: They are all zero Explanation

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5. Suppose A is a real symmetric matrix and λ is an eigenvalue of A with eigenvector x. Which of the following is always true?

a: x is complex Explanation

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b: λ is complex Explanation

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c: λ is real and x is real Explanation

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d: λ is real and x can be chosen to be real Explanation

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6. Let A be a diagonalizable matrix with a repeated eigenvalue λ. Which of the following is true?

a: The geometric multiplicity of λ must be 1 Explanation

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b: The geometric multiplicity of λ equals its algebraic multiplicity Explanation

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c: The algebraic multiplicity of λ is 1 Explanation

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d: The matrix must be defective Explanation

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7. Let A be an upper triangular matrix. What can be said about its eigenvalues?

a: They are all zero Explanation

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b: They are all equal Explanation

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c: They are the entries on the main diagonal Explanation

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d: They are the sum of each row Explanation

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8. Suppose A is an n×n matrix such that A is not invertible. Which of the following is necessarily true?

a: All eigenvalues are zero Explanation

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b: A is diagonalizable Explanation

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c: 0 is an eigenvalue of A Explanation

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d: A has full rank Explanation

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9. Let A be an n×n real matrix such that Aᵗ = A⁻¹. What type of matrix is A?

a: Symmetric Explanation

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b: Orthogonal Explanation

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c: Diagonal Explanation

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d: Skew-symmetric Explanation

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10. Let A be a non-diagonalizable n×n matrix. Which of the following must be true?

a: A is defective Explanation

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b: All eigenvalues are distinct Explanation

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c: A is symmetric Explanation

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d: A is invertible Explanation

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11. Let A be an n×n matrix such that A^k = 0 for some positive integer k. What can be said about the minimal polynomial of A?

a: It must be x Explanation

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b: It must divide x^k Explanation

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c: It must be irreducible Explanation

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d: It must be of degree n Explanation

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12. Let V be a vector space of dimension n and let T: V → V be a linear operator such that T² = T. Which of the following is necessarily true?

a: T is the identity operator Explanation

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b: T is nilpotent Explanation

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c: T is diagonalizable Explanation

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d: T is invertible Explanation

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13. Which condition is necessary for a square matrix A to be diagonalizable?

a: A must be symmetric Explanation

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b: A must have distinct eigenvalues Explanation

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c: The geometric multiplicity of each eigenvalue equals its algebraic multiplicity Explanation

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d: A must be invertible Explanation

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14. Let T: ℝⁿ → ℝⁿ be a linear operator such that T has no non-trivial invariant subspaces. Which of the following must be true?

a: T is a scalar multiple of the identity Explanation

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b: T is not diagonalizable Explanation

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c: T is irreducible and acts like a cyclic shift Explanation

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d: T is the zero map Explanation

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15. Let A be a square matrix. If A is diagonalizable, which of the following is true?

a: There exists an invertible matrix P such that A = PDP⁻¹ Explanation

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b: A is always symmetric Explanation

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c: A has only one eigenvalue Explanation

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d: A must be orthogonal Explanation

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16. Let A be a 4×4 real matrix such that A^T = -A. What can be said about the eigenvalues of A?

a: They are all real Explanation

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b: They are all purely imaginary or zero Explanation

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c: They are all 1 or -1 Explanation

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d: They are all non-negative Explanation

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17. If A and B are n×n matrices such that AB = BA, which of the following is always true?

a: A and B are both diagonalizable Explanation

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b: A and B are similar Explanation

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c: AB is symmetric Explanation

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d: A and B share a common eigenvector (if over ℂ) Explanation

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18. Let V be an inner product space and T: V → V be a linear operator such that ⟨T(x), y⟩ = ⟨x, T(y)⟩ for all x, y ∈ V. Which of the following must be true?

a: T is self-adjoint (Hermitian) Explanation

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b: T is unitary Explanation

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c: T is nilpotent Explanation

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d: T is skew-symmetric Explanation

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19. Suppose A is a real matrix such that A^T A = A A^T. Which of the following is necessarily true?

a: A is symmetric Explanation

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b: A is diagonalizable Explanation

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c: A is normal Explanation

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d: A is orthogonal Explanation

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20. If A is a square matrix and A² = 0, which of the following must be true?

a: A is invertible Explanation

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b: A has full rank Explanation

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c: All eigenvalues of A are zero Explanation

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d: A is a diagonal matrix Explanation

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Understand various matrix operations, matrix decompositions, factorization and related operations