Exercise6

There is a relationship between the determinant of a matrix and the product of its eigenvalues.

  1. We have seen that the eigenvalues of the matrix \(A = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 1 \\ \end{array}\right]\) are \(\lambda = 3,-1\text{.}\) What is \(\det A\text{?}\) What is the product of the eigenvalues of \(A\text{?}\)

  2. Consider the triangular matrix \(A = \left[\begin{array}{rrr} 2 \amp 0 \amp 0 \\ -1 \amp -3 \amp 0 \\ 3 \amp 1 \amp -2 \\ \end{array}\right] \text{.}\) What are the eigenvalues of \(A\text{?}\) What is \(\det A\text{?}\) What is the product of the eigenvalues of \(A\text{?}\)

  3. Based on these examples, what do you think is the relationship between the determinant of a matrix and the product of its eigenvalues?

  4. Suppose the characteristic polynomial is written as

    \begin{equation*} \det(A-\lambda I) = (\lambda_1-\lambda)(\lambda_2-\lambda) \ldots (\lambda_n-\lambda) \text{.} \end{equation*}

    By substituting \(\lambda = 0\) into this equation, explain why the determinant of a matrix equals the product of its eigenvalues.

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