Virtual Labs

The rank of A =

2  -3  1
          1  -2  5
          -4  6  -7

is:

a: 1 Explanation

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b: 2 Explanation

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c: 3 Explanation

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d: 0 Explanation

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The minor M(1,3) of A =

1  -2  5
          -4  2  -3
          -5  3  -2

is:

a: -2 Explanation

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b: -5 Explanation

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c: -8 Explanation

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d: 2 Explanation

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The cofactor C(2,1) of A =

3  -2  1
          6  -1  -5
          -2  4  -3

is:

a: -2 Explanation

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b: 10 Explanation

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c: 6 Explanation

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d: -6 Explanation

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The minor M(3,2) of A =

3  -2  -3
          -4  1  6
          1  -2  3

is:

a: 4 Explanation

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b: 6 Explanation

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c: -6 Explanation

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d: -4 Explanation

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Eigenvalues of A =

2  3
          2  1

are:

a: 4, -1 Explanation

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b: 2, 3 Explanation

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c: 1, -2 Explanation

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d: 3, -3 Explanation

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Eigenvectors of A =

0  1
          -2  -3

are:

1  -2

and

1  -1

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1  1

and

1  -1

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1  2

and

1  -3

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1  3

and

1  -2

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Eigenvalues of a Hermitian matrix are:

a: Real but not necessarily positive Explanation

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b: Always positive Explanation

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c: Purely imaginary Explanation

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d: Complex Explanation

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The conjugate transpose of matrix A =

1+2j  5-2j  -6+2j
          -1-3j  -1-5j  -3-j
          2+3j  -2-4j  2+j

is:

1-2j  -1+3j  2-3j
          5+2j  -1+5j  -2+4j
          -6-2j  -3+j  2-j

Explanation

1+2j  -1-3j  2+3j
          5-2j  -1-5j  -2-4j
          -6+2j  -3-j  2+j

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1+2j  5-2j  -6+2j
          -1-3j  -1-5j  -3-j
          2+3j  -2-4j  2+j

Explanation

1-2j  5+2j  -6-2j
          -1+3j  -1+5j  -3+j
          2-3j  -2+4j  2-j

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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