Comment. Matrix quiz
Comment. This quiz is designed to test your knowledge of such concepts as diagonalizability,
Comment. invertibility, row operations, rank, determinant, etc.
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Question. Let A be an n x n matrix. Which of the following criteria will ensure that A is diagonalizable over the reals?
Answer. The rows of A are linearly independent.
Comment. This ensures that A is invertible, but invertibility has nothing to do with diagonalizability.
Answer. The characteristic polynomial of A splits over the reals.
Comment. This is not quite enough, because if there are repeated roots it can still happen that A is not diagonalizable.
Answer. The determinant of A is non-zero.
Comment. This ensures that A is invertible, but invertibility has nothing to do with diagonalizability.
Correct Answer. A has n distinct real eigenvalues.
Partially Correct Answer. The characteristic polynomial of A has no repeated roots.
Comment. 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.
Answer. A can be row reduced to a diagonal matrix.
Comment. Row reduction does not necessarily preserve diagonalizability.
Answer. A commutes with all diagonal matrices.
Question. Let A be an n x n invertible matrix with real entries. Which of the following statements is NOT necessarily true?
Answer. A can be written as the product of elementary matrices.
Comment. If A is invertible, then it has rank n, and so A can be row reduced to the identity.
Answer. det(A) is non-zero.
Comment. If A is invertible then det(A) det(A^{-1}) = det(I) = 1, and hence det(A) is non-zero.
Answer. Zero is not an eigenvalue of A.
Comment. If zero was an eigenvalue then the null space of A would contain a non-zero vector, and A would not be invertible.
Correct Answer. A is similar to the identity matrix.
Comment. The identity matrix is not similar to any matrix other than itself.
Answer. The linear transformation x -> Ax, from R^n to R^n, is both one-to-one and onto.
Answer. One can row reduce A to the identity matrix.
Answer. The null space of A contains only the zero vector.
Question. Let A be an invertible 5 x 5 matrix. Which of the following statements is _false_?
Answer. The rank of A must equal 5.
Correct Answer. Every row of A must contain a leading 1.
Answer. For every vector b in R^5, there must be exactly one solution to the equation Ax = b.
Answer. The reduced row echelon form of A must be the identity matrix.
Answer. The row-reduced echelon form of A must contain no free variables.
Answer. The linear transformation associated to A must be both one-to-one and onto.
Answer. There must exist a 5 x 5 matrix B, such that AB = BA = I.
Question. Let A be an invertible 5 x 5 matrix. Which of the following statements is _false_?
Answer. The image of A is R^5.
Correct Answer. There are five distinct eigenvalues.
Answer. The columns of A form a basis for R^5.
Answer. The rows of A form a basis for R^5.
Answer. All the eigenvalues of A are non-zero.
Answer. The determinant of A is non-zero.
Answer. The kernel of A is {0}.
Question. If one applies row reduction to a matrix, then
Correct Answer. The image may change, but the kernel, rank, and nullity do not change.
Answer. The image, kernel, rank, and nullity all do not change.
Answer. The kernel may change, but the image, rank, and nullity do not change.
Answer. The image, kernel, and nullity may change, but the rank does not change.
Answer. The image, kernel, rank, and nullity may all change.
Answer. The image, rank, and kernel may change, but the nullity does not change.
Answer. The image and kernel may change, but the rank and nullity do not change.
Question. If one replaces a matrix with its transpose, then
Answer. The image may change, but the kernel, rank, and nullity do not change.
Answer. The image, kernel, rank, and nullity all do not change.
Answer. The kernel may change, but the image, rank, and nullity do not change.
Correct Answer. The image, kernel, and nullity may change, but the rank does not change.
Answer. The image, kernel, rank, and nullity may all change.
Answer. The image, rank, and kernel may change, but the nullity does not change.
Answer. The image and kernel may change, but the rank and nullity do not change.
Comment. This is only true if the matrix is square.
Question. Let A be a diagonalizable 5 x 5 matrix. Which one of the following statements is _false_?
Correct Answer. All the eigenvalues must be distinct (occur with algebraic and geometric multiplicity 1).
Comment. This will imply diagonalizability but is not implied by it.
Answer. There must exist a basis of R^5 consisting entirely of eigenvectors of A.
Answer. The algebraic multiplicity of each eigenvalue must equal the geometric multiplicity of each eigenvalue
Answer. The total algebraic multiplicities of all the eigenvalues must equal 5.
Answer. The total geometric multiplicities of all the eigenvalues must equal 5.
Answer. The determinant of A must equal the product of all the eigenvalues (counted with multiplicity)
Answer. The rank of A is equal to the number of non-zero eigenvalues (counted with multiplicity)