Exercise4

Provide a justification for your response to the following questions.

  1. If \(A\) is a \(3\times3\) matrix having eigenvalues \(\lambda = 2, 3, -4\text{,}\) can you guarantee that \(A\) is diagonalizable?

  2. If \(A\) is a \(2\times 2\) matrix with a complex eigenvalue, can you guarantee that \(A\) is diagonalizable?

  3. If \(A\) is similar to the matrix \(B = \left[\begin{array}{rrr} -5 \amp 0 \amp 0 \\ 0 \amp -5 \amp 0 \\ 0 \amp 0 \amp 3 \\ \end{array}\right] \text{,}\) is \(A\) diagonalizable?

  4. What matrices are similar to the identity matrix?

  5. If \(A\) is a diagonalizable \(2\times2\) matrix with a single eigenvalue \(\lambda = 4\text{,}\) what is \(A\text{?}\)

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