Hidden structure

Compare the rows with each other before computing anything.

1. Every row is a multiple of $(4,0,1)$, so $A$ has rank $1$ and can be written as $A=u{v}^T$ with $u=(1,2,3)$ and $v=(4,0,1)$.
2. Every vector orthogonal to $v$ is sent to $0$, which gives a plane of eigenvectors with eigenvalue $0$.
3. Also $Au=u(v^{T}u)$, so $u$ is an eigenvector with eigenvalue $v^{T}u=4+0+3=7$, which matches the trace.
4. So the largest eigenvalue is $7$.

Answer: $7$