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Quiz: Linear algebra
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last edited
by RH 15 years, 3 months ago
This quiz tests your understanding of such concepts as diagonalizability, invertibility, row operations, determinant, etc.
Discuss this quiz
(Key: correct, incorrect, partially correct.)
- Let A be an n x n matrix. Which of the following criteria will ensure that A is diagonalizable over the reals?
- The rows of A are linearly independent.
- INCORRECT. This ensures that A is invertible, but invertibility has nothing to do with diagonalizability.
- The characteristic polynomial of A splits over the reals.
- INCORRECT. This is not quite enough, because if there are repeated roots it can still happen that A is not diagonalizable.
- The determinant of A is non-zero.
- INCORRECT. This ensures that A is invertible, but invertibility has nothing to do with diagonalizability.
- A has n distinct real eigenvalues.
- CORRECT. Note, though, that it is possible for A to be diagonalizable while also having repeated eigenvalues.
- The characteristic polynomial of A has no repeated roots.
- PARTIALLY. This is pretty close. However, if the characteristic polynomial doesn't split, e.g. if it is lambda^2 + 1, then A is still not diagonalizable.
- A can be row reduced to a diagonal matrix.
- INCORRECT. Row reduction does not necessarily preserve diagonalizability.
- Let A be an invertible 5 x 5 matrix. Which of the following statements is false?
- The rank of A must equal 5.
- Every row of A must contain a leading 1.
- For every vector b in , there must be exactly one solution to the equation Ax = b.
- The reduced row echelon form of A must be the identity matrix.
- The row-reduced echelon form of A must contain no free variables.
- The linear transformation associated to A must be both one-to-one and onto.
- There must exist a 5 x 5 matrix B, such that AB = BA = I.
- Let A be an invertible 5 x 5 matrix. Which of the following statements is false?
- The image of A is .
- There are five distinct eigenvalues.
- The columns of A form a basis for .
- The rows of A form a basis for .
- All the eigenvalues of A are non-zero.
- The determinant of A is non-zero. The kernel of A is {0}.
- Let A be an n x n invertible matrix with real entries. Which of the following statements is NOT necessarily true?
- A can be written as the product of elementary matrices.
- INCORRECT. If A is invertible, then it has rank n, and so A can be row reduced to the identity.
- det(A) is non-zero.
- INCORRECT. If A is invertible then det(A) det() = det(I) = 1, and hence det(A) is non-zero.
- Zero is not an eigenvalue of A.
- INCORRECT. If zero was an eigenvalue then the null space of A would contain a non-zero vector, and A would not be invertible.
- A 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 A to the identity matrix.
- The null space of A contains only the zero vector.
- 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 5 x 5 matrix. Which one of the following statements is false?
- All the eigenvalues must be distinct (i.e. 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 A.
- 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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Quiz: Linear algebra
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