T/F Midterm 3-Linear Algebra

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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)


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