# The term $ae + ai + ei - hf - bd - cg$ in characteristic equation of $3 \times 3$ matrix has any special name?

For $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ characteristic equation is given by $\lambda^2-tr(A) + |A|=0$. Where $tr(A)$ and $|A|$ are trace and determinant of a matrix A respectively. I was wondering what it would be for $3 \times 3$ matrix $A = \begin{bmatrix} a & b & c\\ d & e & f\\g & h & i \end{bmatrix}$?

On solving it for its characteristic equation as $|A - \lambda I| = 0$

I got in the form \begin{equation} -\lambda^3+tr(A)\lambda^2-(ae + ai + ei - hf - bd - cg)\lambda + |A|=0. \end{equation} I was wondering what can we call that coefficient term of $\lambda$. Does it have any special name.

## 1 Answer

No, it doesn't have a name, but it is $\operatorname{tr}(\bigwedge^2 A)$, or equivalently the sum of principal $2\times 2$ minors, or, in terms of trace and powers of $A$, as $\frac12[\operatorname{tr}(A)^2-\operatorname{tr}(A^2)]$).

• What does $\bigwedge^2$ represents – Meet Sep 19 '18 at 15:42
• Second exterior power. $\bigwedge^2 A$ is the linear map $v\wedge w\mapsto (Av)\wedge(Aw)$. If you haven't seen it before, don't worry too much about it. – user10354138 Sep 19 '18 at 16:36