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