Could someone please explain why following is correct?
$$b+Ch=0$$ $$h=bC^{-1}$$
Note that $b$ is the gradient and $C$ the hessian.
I'm not familiar with algebraic transformations of vector functions. It seems like there are different rules to apply. I would be also glad to know a good resource to learn more about this topic.
If $b$ and $h$ are column vectors, and $C$ is a nonsingular $(n\times n)$-matrix then $b+Ch=0$ implies $$h=-C^{-1}b\ .$$ If $C$ is the Hessian of some $C^2$-function $f:\>{\mathbb R}^n\to{\mathbb R}$ at some point $p$ then $C$ is a symmetric matrix. It follows that we also have $$h'=- b'C^{-1}\ ,$$ whereby $h'$ and $b'$ now are row vectors.
First thing next week should be for you to take in some linear algebra!