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Quiz: Matrices

Page history last edited by RH 12 years, 1 month ago

TThis quiz is designed to test your knowledge of such concepts as diagonalizability, invertibility, row operations, rank, determinant, etc.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. Let Formula be an Formula matrix. Which of the following criteria will ensure that Formula is diagonalizable over the reals?
    1. The rows of Formula are linearly independent.
      • INCORRECT. This ensures that Formula is invertible, but invertibility has nothing to do with diagonalizability.
    2. The characteristic polynomial of Formula splits over the reals.
      • INCORRECT. This is not quite enough, because if there are repeated roots it can still happen that Formula is not diagonalizable.
    3. The determinant of Formula is non-zero.
      • INCORRECT. This ensures that Formula is invertible, but invertibility has nothing to do with diagonalizability.
    4. Formula has Formula distinct real eigenvalues.
      • CORRECT.
    5. The characteristic polynomial of Formula has no repeated roots.
      • PARTIALLY. This is pretty close. However, if the characteristic polynomial doesn't split, e.g. if it is Formula, then Formula is still not diagonalizable.
    6. Formula can be row reduced to a diagonal matrix.
      • INCORRECT. Row reduction does not necessarily preserve diagonalizability.
    7. Formula commutes with all diagonal matrices.
  2. Let Formula be an Formula invertible matrix with real entries. Which of the following statements is not necessarily true?
    1. Formula can be written as the product of elementary matrices.
      • INCORRECT. If Formula is invertible, then it has rank Formula, and so Formula can be row reduced to the identity.
    2. Formula is non-zero.
      • INCORRECT. If Formula is invertible then Formula, and hence Formula is non-zero.
    3. Zero is not an eigenvalue of Formula.
      • INCORRECT. If zero was an eigenvalue then the null space of Formula would contain a non-zero vector, and Formula would not be invertible.
    4. Formula is similar to the identity matrix.
      • CORRECT. The identity matrix is not similar to any matrix other than itself.
    5. The linear transformation Formula, from Formula to Formula, is both one-to-one and onto.
    6. One can row reduce Formula to the identity matrix.
    7. The null space of Formula contains only the zero vector.
  3. Let Formula be an invertible Formula matrix. Which of the following statements is false?
    1. The rank of Formula must equal 5.
    2. Every row of Formula must contain a leading 1.
      • CORRECT.
    3. For every vector Formula in Formula, there must be exactly one solution to the equation Formula.
    4. The reduced row echelon form of Formula must be the identity matrix.
    5. The row-reduced echelon form of Formula must contain no free variables.
    6. The linear transformation associated to Formula must be both one-to-one and onto.
    7. There must exist a Formula matrix Formula, such that Formula.
  4. Let Formula be an invertible Formula matrix. Which of the following statements is false?
    1. The image of Formula is Formula.
    2. There are five distinct eigenvalues.
      • CORRECT.
    3. The columns of Formula form a basis for Formula.
    4. The rows of Formula form a basis for Formula.
    5. All the eigenvalues of Formula are non-zero.
    6. The determinant of Formula is non-zero.
    7. The kernel of Formula is Formula.
  5. If one applies row reduction to a matrix, then
    1. The image may change, but the kernel, rank, and nullity do not change.
      • CORRECT.
    2. The image, kernel, rank, and nullity all do not change.
    3. The kernel may change, but the image, rank, and nullity do not change.
    4. The image, kernel, and nullity may change, but the rank does not change.
    5. The image, kernel, rank, and nullity may all change.
    6. The image, rank, and kernel may change, but the nullity does not change.
    7. The image and kernel may change, but the rank and nullity do not change.
  6. If one replaces a matrix with its transpose, then
    1. The image may change, but the kernel, rank, and nullity do not change.
    2. The image, kernel, rank, and nullity all do not change.
    3. The kernel may change, but the image, rank, and nullity do not change.
    4. The image, kernel, and nullity may change, but the rank does not change.
      • CORRECT.
    5. The image, kernel, rank, and nullity may all change.
    6. The image, rank, and kernel may change, but the nullity does not change.
    7. The image and kernel may change, but the rank and nullity do not change.
      • INCORRECT. This is only true if the matrix is square.
  7. Let A be a diagonalizable Formula matrix. Which one of the following statements is false?
    1. All the eigenvalues must be distinct (occur with algebraic and geometric multiplicity 1).
      • CORRECT. This will imply diagonalizability but is not implied by it.
    2. There must exist a basis of Formula consisting entirely of eigenvectors of Formula.
    3. The algebraic multiplicity of each eigenvalue must equal the geometric multiplicity of each eigenvalue
    4. The total algebraic multiplicities of all the eigenvalues must equal 5.
    5. The total geometric multiplicities of all the eigenvalues must equal 5.
    6. The determinant of A must equal the product of all the eigenvalues (counted with multiplicity)
    7. The rank of A is equal to the number of non-zero eigenvalues (counted with multiplicity)

 

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