This is when studying about Chern classes from Kobayashi and Nomizu.

Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$.

Let $p:P\rightarrow M$ be associated principal $G$ bundle. Let $\mathfrak{g}=\mathfrak{gl}(r,\mathbb{C})$ denote the Lie algebra of $G$.

Given $B\in \mathfrak{g}$, the determinant $\text{det}\left(\lambda I_r-\frac{1}{2\pi\sqrt{-1}} B\right)$ is $\sum_{k=0}^r a_k\lambda^{r-k}$ for some $a_k\in \mathbb{C}$.

Given $B\in \mathfrak{g}$, we have $r$ elements in $\mathbb{C}$. Varying $B$ over $\mathfrak{g}$, gives $r$ functions $f_k:\mathfrak{g}\rightarrow \mathbb{C}$.

We have $$\text{det}\left(\lambda I_r-\frac{1}{2\pi\sqrt{-1}} X\right)=\sum_{k=0}^r f_k(X) \lambda ^{r-k}$$ These $f_k:\mathfrak{gl}(r,\mathbb{C})\rightarrow \mathbb{C}$ are homogeneous, degree $k$ polynomial functions on $\mathfrak{gl}(r,\mathbb{C})$. I can recall what are polynomial functions on a vector space if some one needs it.

These are $GL(r,\mathbb{C})$ invariant i.e., $f_k(X)=f_k(DXD^{-1})$ for all $D\in Gl(r,\mathbb{C})$. These $f_k$ gives a symmetric, multilinear, $Gl(r,\mathbb{C})$ invariant $\mathbb{C}$ valued mappings $f_k\in I_{\mathbb{C}}^k(G)$.

Let $\Gamma$ be a connection on $P(M,G)$ and $\Omega$ be its curvature form. This $f_k$ gives a $\mathbb{C}$ valued $2k$ form $f_k(\Omega)$ on $P$. Let $\gamma_k$ be the unique $\mathbb{C}$ valued closed $2k$-form on $M$ such that $p^*(\gamma_k)=f_k(\Omega)$.

This gives $[\gamma_k]\in H^{2k}(M,\mathbb{C})$. But $k$-th Chern class are supposed to take values in $H^{2k}(M,\mathbb{R})$. What is the obvious map $H^{2k}(M,\mathbb{C})\rightarrow H^{2k}(M,\mathbb{R})$ to look for to get an element in $H^{2k}(M,\mathbb{R})$?

We then have $$\sum_{k=0}^r f_k(\Omega)=\sum_{k=0}^rp^*(\gamma_k)=p^*(1+\gamma_1+\cdots+\gamma_r)$$ Then,they (Kobayashi and Nomizu,page no $307$) write $$\text{det}\left(I_r-\frac{1}{2\pi \sqrt{-1}} \Omega\right)= p^*(1+\gamma_1+\cdots+\gamma_r)$$ I see that they are just replacing $X$ with $\Omega$. But what does it mean to say determinant of $\Omega$?

We have $\gamma_k\in H^{2k}(M,\mathbb{R})$. So, $1+\gamma_1+\cdots+\gamma_r\in H^*(M,\mathbb{R})$ which then imply that $p^*(1+\gamma_1+\cdots+\gamma_r)\in H^*(P,\mathbb{R})$. So, $\text{det}\left(I_r-\frac{1}{2\pi i} \Omega\right)\in H^*(P,\mathbb{R})$. How is it defined?