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