Exercise6
There is a relationship between the determinant of a matrix and the product of its eigenvalues.
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 \(A\text{?}\)
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 \(A\text{?}\)
Based on these examples, what do you think is the relationship between the determinant of a matrix and the product of its eigenvalues?

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.