Understanding Determinant Equations for Linear Algebra | Homework Help

[tylerc1991]

Homework Statement

On my professor's study guide, he mentions that for the upcoming test we need to be able to solve determinant equations. What does this mean? I have tried to google what a determinant equation is and can't find anything, so am I safe in a assuming that he means that we need to be able to solve a linear system USING determinants (i.e. Cramer's rule)? Or could someone enlighten me as to what a determinant equation is? Thank you!

 

[HallsofIvy]

A "determinant equation" is, of course, an equation that involves a determinant. I presume that the determinant will depend upon some unknown. For example, to find an eigenvalue for, say, the matrix
$$\begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 2\\ 3 & 2 & 1\end{bmatrix}$$
you would have to solve the equation
$$\left|\begin{array}{ccc}1- \lambda & 2 & 1 \\ 0 & 1- \lambda & 2 \\ 3 & 2 & 1- \lambda\end{array}\right|= 0$$
a "determinant equation". Evaluating the determinant gives you a cubic equation to solve for $\lambda$.