Exercise 6

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 $A = \left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right]$ are $\lambda = 3,-1$. What is $\det A$? What is the product of the eigenvalues of $A$?

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

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 $$\det(A-\lambda I) = (\lambda_1-\lambda)(\lambda_2-\lambda) \ldots (\lambda_n-\lambda)$$. By substituting $\lambda = 0$ into this equation, explain why the determinant of a matrix equals the product of its eigenvalues.