Exercise4

Provide a justification for your response to the following questions.

  1. Suppose that \(A\) is a \(3\times 3\) matrix having eigenvalues \(\lambda = -3,3,-5\text{.}\) What are the eigenvalues of \(2A\text{?}\)

  2. Suppose that \(D\) is a diagonal \(3\times 3\) matrix. Why can you guarantee that there is a basis of \(\real^3\) consisting of eigenvectors of \(D\text{?}\)

  3. If \(A\) is a \(3\times 3\) matrix whose eigenvalues are \(\lambda = -1,3,5\text{,}\) can you guarantee that there is a basis of \(\real^3\) consisting of eigenvectors of \(A\text{?}\)

  4. Suppose that the characteristic polynomial of a matrix \(A\) is

    \begin{equation*} \det(A-\lambda I) = -\lambda^3 + 4\lambda \text{.} \end{equation*}

    What are the eigenvalues of \(A\text{?}\) Is \(A\) invertible? Is there a basis of \(\real^n\) consisting of eigenvectors of \(A\text{?}\)

  5. If the characteristic polynomial of \(A\) is

    \begin{equation*} \det(A-\lambda I) = (4 -\lambda)(-2-\lambda)(1-\lambda) \text{,} \end{equation*}

    what is the characteristic polynomial of \(A^2\text{?}\) what is the characteristic polynomial of \(A^{-1}\text{?}\)

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