Order of equation that is used to get the matrix for calculating eigenvalues?

Question

My professor in class has taught this equation as being $det(\lambda I - A)$ mostly with regards to finding the eigenvalues and eigenvectors of liner transformations but I have seen it online as $det(A - \lambda I)$. Does the order matter in these cases? I tried to google this but I couldn't find anything

Answer 1

It doesn't matter. Recall that multiplying a single row (or equivalently, column) of a matrix $M$ by a scalar $a$ multiplies the determinant by $a$. Note also that $\lambda I - A = (-1)(A - \lambda I)$ is the result of multiplying all rows (or columns) by $-1$. Therefore we have, $$\det (A - \lambda I) = (-1)^n\det(\lambda I - A)$$ where $n$ is the number of rows/columns.

So, we therefore have \begin{align*} &\det (A - \lambda I) = 0 \\ \iff \, &(-1)^n\det(\lambda I - A) = 0 \\ \iff \, &\det(\lambda I - A) = 0. \end{align*}