Ch 5 True/False

(5.2) The determinant of A is the product of the diagonal entries in A.

False in general. True if A is triangular.

(5.2) A row replacement operation on A does not change the eigenvalues.

False.

(5.2) If λ + 5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.

False. -5 is an eigenvalue.

(5.3) A is diagonalizable if A = PDP^-1 for some matrix D and some invertible matrix P.

False. D must be a diagonal matrix.

(5.2) An elementary row operation on A does not change the determinant.

False. Interchanging rows and multiplying a row by a constant changes the determinant.

(5.3) A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.

False. It always has n eigenvalues, counting multiplicity.

(5.3) If A is diagonalizable, then A has n distinct eigenvalues.

False. It could have repeated eigenvalues as long as the basis of each eigenspace is equal to the multiplicity of that eigenvalue. The converse is true however.

(5.3) If A is diagonalizable, then A is invertible.

False. It's invertible if it doesn't have zero as an eigenvector but this doesn't affect diagonalizabilty.

(5.2) If A is 3x3, with columns a1, a2 and a3, then detA equals the volume of the parallelpiped determined by a1, a2 and a3.

False. It's the absolute value of the determinant.

(5.1) To find the eigenvalues of A, reduce A to echelon form.

False. Row reducing changes the eigenvectors and eigenvalues.

(5.1) If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues.

False. The converse is true, however.

(5.3) A is diagonalizable if A has n eigenvectors.

False. The eigenvectors have to be be linearly independent.

(5.1) If Ax = λx for some scalar, then x is an eigenvector of A.

False. The vector must be nonzero.

(5.3) If A is invertible, then A is diagonalizable.

False. These are not directly related.

(5.1) The eigenvalues of a matrix are on its main diagonal.

False. This is only true for triangular matrices.

(5.1) If Ax = λx for some vector x, then λ is an eigenvalue of A.

False. This is true as long as the vector is not the zero vector.

(5.2) det(A^T) = (-1)detA

False. det(A^T) = detA

(5.1) A matrix A is not invertible if and only if 0 is an eigenvalue of A.

True.

(5.2) (detA)(detB) = detAB

True.

(5.3) If R^n has a basis of eigenvectors of A, then A is diagonalizable.

True.

(5.1) A steady-state vector for a stochastic matrix is actually an eigenvector.

True. A steady state vector has the property that Ax = x. In this case λ is 1.

(5.3) If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.

True. Each column of PD is a column of P times A and is equal to the corresponding entry in D times the vector P. This satisfies the eigenvector definition as long as the column is nonzero.

(5.1) Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.

True. Just see if Ax is a scalar multiple of x.

(5.2) The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A.

True. That's the definition.

(5.1) An eigenspace of A is a null space of a certain matrix.

True. The eigenspace is the nullspace of A − λI.

(5.1) A number c is an eigenvalue of A if and only if the equation (A - cI)x=0 has a nontrivial solution.

True. This is a rearrangement of the equation Ax = λx