What is the difference between $\det(zI - A)$ and $\det(A - zI)$?

Question

When computing the characteristic polynomial of matrix $A$, is it the case that $$\det(zI - A) = \det(A - zI)$$ and that one should choose the construction that is best on a case-by-case basis? If so, is one of these preferred over the other? If they are not equal, which one represents the characteristic polynomial of matrix $A$?

Answer 1

Answering my own question (with reference to the great comments that were left):

It is the case that $\det(zI - A)$ is not necessarily equal to $\det(A - zI)$. Specifically, one can obtain matrix $A - zI$ from matrix $zI - A$ by pulling a factor of $-1$ out of each row of matrix $zI - A$. Therefore, $\det(zI - A) = (-1)^n \det (A - zI)$. As a result, when $n$ is odd, $\det(zI - A) \neq \det(A - zI)$.

To determine which form correctly represents the characteristic polynomial of $A$, consider that $\det(A - zI) = (\lambda_1 - z)\cdots(\lambda_n - z)$ where $\lambda_1, \dots, \lambda_n$ are the eigenvalues of $A$, repeated per their algebraic multiplicities. Hence, you can see that the leading term of $\det(A - zI)$ will be $(-1)^n z^n$, which will be negative if $n$ is odd. So, the form $\det(A - zI)$ will not produce a monic polynomial when $n$ is odd, and therefore, $\det(A - zI)$ should not be used to represent the characteristic polynomial. By contrast, $\det(zI - A) = (z - \lambda_1) \cdots (z - \lambda_n)$ so that the leading term is $z^n$, and the polynomial produced from $\det(zI - A)$ is monic, and therefore $\det(zI - a)$ correctly represents the characteristic polynomial of $A$.

All that said, the most common use of the characteristic polynomial is to find the eigenvalues of $A$. And for this application, recognize that the roots of $\det(zI - A)$ are the same as the roots of $\det(A - zI)$, these roots being the eigenvalues of $A$. Therefore, when computing the eigenvalues of $A$, the equation $\det(A - \lambda I) = 0$ will produce the same eigenvalues as the equation $\det(\lambda I - A) = 0$. Moreover, when computing these eigenvalues by hand, the matrix $A - \lambda I$ requires fewer modifications to the original matrix $A$, and therefore is less likely to result in bookkeeping errors (i.e. dropping a minus sign). So, the equation $\det(A - \lambda I) = 0$ is actually preferred when doing hand computations.

To summarize, you will often see $\det(A - \lambda I) = 0$ used for computing eigenvalues. But if you want to correctly represent the characteristic polynomial, you should write $\det(zI - A)$.