Math 511: Linear Algebra

True/False Chapters 6 and 7

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True/False

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For each of the following exercises, if the statement is true indicate that it is true by showing or proving that the statement is true. If the statement is false, create a counter example illustrating that the statement is false.

1. If $A$ is an $n\times n$ matrix, then whose eigenvalues are all nonzero, then $A$ is nonsingular.

2. If $A$ is an $n\times n$ matrix, then $A$ and $A^T$ have the same eigenvectors.

3. If $A$ and $B$ are similar $n\times n$ matrices, then $A$ and $B$ have the same eigenvalues.

4. If $A$ and $B$ are $n\times n$ matrices with the same eigenvalues, then they are similar.

5. If $A$ is an $n\times n$ matrix that has eigenvalues of algebraic multiplicity greater than one, then $A$ is defective.

6. If $A$ is a $4\times 4$ matrix with rank 3, and $\lambda=0$ is an eigenvalue with algebraic multiplicity 3, then $A$ is diagonalizable.

7. If $A$ is a $4\times 4$ matrix with rank 1, and $\lambda=0$ is an eigenvalue with multiplicity 3, then $A$ is defective.

8. The rank of an $n\times n$ matrix $A$ is equal to the number of nonzero eigenvalues of $A$, where eigenvalues are counted according to algebraic multiplicity.

9. The rank of an $m\times n$ matrix $A$ is equal to the number of nonzero singular values of $A$, where singular values are counted according to multiplicity.

10. If $A$ is Hermitian and $c\in\mathbb{C}$ is a complex number, then $cA$ is Hermitian.