###### 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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