T/F Midterm 3-Linear Algebra
Suppose A is an n × n matrix. We say that a real number λ is an eigenvalue of A if...
Av = λv has a non-trivial solution. (equivalently: Av = λv for some non-zero v in R n , etc.)
A 5×5 real matrix has an even number of real eigenvalues.
False
A is diagonalizable if and only if A has n eigenvalues, counting multiplicity.
False
If A and B are 3×3 matrices that have the same eigenvalues and the same algebraic multiplicity for each eigenvalue, then A = B.
False
If A and B are nxn matrices with det(A) = 0 and det(B) = 0, then det(A+B) = 0.
False
If A and B have the same eigenvectors, then A and B have the same characteristic polynomial.
False
If A is a 3×3 matrix with characteristic polynomial −λ 3 + λ 2 + λ, then A is invertible.
False
If A is an n × n matrix and its eigenvectors form a basis for R n , then A is invertible.
False
If A is an n × n matrix and λ = 2 is an eigenvalue of A, then Nul(A− 2I) = {0}.
False
If A is an n × n matrix then det(−A) = −det(A).
False
If A is an nxn matrix and det(A) = 2, then 2 is an eigenvalue of A.
False
If A is an nxn matrix and v and w are eigenvectors of A, then v + w is also an eigenvector of A.
False
If A is an nxn matrix, then the determinant of A is the same as the determinant of the RREF of A.
False
If A is diagonalizable, then A has n distinct eigenvalues.
False
If A is diagonalizable, then A is invertible.
False
If A is row equivalent to B, then A and B have the same eigenvalues.
False
If A is similar to B, then A and B have the same eigenvectors.
False
If Ais a 2×2 matrix and det(A) = det(−A), then Ais not invertible.
False
If an n×n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.
False
It is possible for a lower-triangular matrix A to have a non-real complex eigenvalue.
False
There exists a real 2×2 matrix with the eigenvalues ii and 2i2i.
False
A 3 × 3 matrix with (only) two distinct eigenvalues is diagonalizable.
Maybe
A diagonalizable n × n matrix admits n linearly independent eigenvectors.
True
A real eigenvalue of a real matrix always has at least one corresponding real eigenvector.
True
Every real 3×3 matrix must have at least one real eigenvalue.
True
If 0 is an eigenvalue of the n × n matrix A, then rank(A) < n.
True
If A is a 3 × 3 matrix and Ae1 = Ae3 , then det(A) = 0.
True
If A is a 3 × 3 matrix with characteristic polynomial (3 − λ) 2 (2 − λ), then the eigenvalue λ = 2 must have geometric multiplicity 1.
True
If A is a square matrix and the homogeneous equation Ax = 0 has only the trivial solution, then A is invertible.
True
If A is an n× n matrix and Nul(A−3I) 6 is not equal to {0}, then λ = 3 must be an eigenvalue of A.
True
If A is diagonalizable and B is similar to A, then B is diagonalizable.
True
If A is diagonalizable, then A^2 is also diagonalizable.
True
If A is the 3x3 matrix satisfying Ae1 = e2 , Ae2 = e3 , and Ae3 = e1 , then det(A) = 1.
True
If det(A) = 0, then 0 is an eigenvalue of A.
True
If there is a basis of Rn consisting of eigenvectors of A, then A is diagonalizable.
True
If v is an eigenvector of a square matrix A, then −v is also an eigenvector of A.
True
Suppose A is an nxn matrix and λ is an eigenvalue of A. If v and w are two different eigenvectors of A corresponding to the eigenvalue λ, then v − w is an eigenvector of A.
True
Suppose a 3 × 3 matrix A has characteristic polynomial −λ^3 + λ. Then A must be diagonalizable.
True
Suppose an n × n matrix A has n linearly independent columns. Then 0 is not an eigenvalue of A.
True
If A is a 3 × 3 matrix with characteristic polynomial det(A− λI) = (1 − λ)(−1 − λ)^2, then A must be invertible.
True (because 0 is not an eigenvalue of A)
If A is a 4 × 4 matrix, then det(−A) = det(A).
True (because it's an even matrix)
