Degree of curve where matrix of polynomials has rank 1 My question is about a step in Exercise 12.8 on page 442 of 3264 & All That by Eisenbud and Harris. Chapter 12 is about Porteous' formula. 
The exercise reads: Let $A=(P_{i,j})$ be a $2 \times 3$ matrix whose entries $P_{i,j}$ are general polynomials of degree $a_{i,j}$ on $\mathbb{P}^3$. Assuming that $a_{1,j}+a_{2,k}=a_{1,k}+a_{2,j}$ for all $j$ and $k$--so that the minors of $A$ are homogeneous--what is the degree of the curve where $A$ has rank $1$?
Shuai Wang posted a solution on page 21 of this document https://www.math.columbia.edu/~tedd2013/intersectiontheory.pdf which seems correct, but I'm struggling to understand one step of it. 
He claims that the curve where $A$ has rank $1$ is the degeneracy locus of the following bundle map: 
$\mathcal{O}_{\mathbb{P}^3}(a_{22}) \oplus \mathcal{O}_{\mathbb{P}^3}(a_{12}) \to \mathcal{O}_{\mathbb{P}^3}(a_{11}+a_{22}) \oplus \mathcal{O}_{\mathbb{P}^3}(a_{12}+a_{22}) \oplus \mathcal{O}_{\mathbb{P}^3}(a_{12}+a_{23})$. 
\begin{align*}
\begin{bmatrix}
    a_{22}  & a_{12}
\end{bmatrix} 
\begin{bmatrix}
    a_{11}       & a_{12} & a_{13} \\
    a_{21}       & a_{22} & a_{23}
\end{bmatrix}
\end{align*}
I have a few questions about this bundle map.  
Why is $\mathcal{O}_{\mathbb{P}^3}(a_{11}+a_{22}) \oplus \mathcal{O}_{\mathbb{P}^3}(a_{12}+a_{22}) \oplus \mathcal{O}_{\mathbb{P}^3}(a_{12}+a_{23})$ the target space of this map?
Why is the degeneracy locus of this map equal the locus of the maximal minors of $A$? The maximal minors of $A$ are $\{a_{11}a_{22}-a_{21}a_{12}, a_{11}a_{23}-a_{13}a_{21}, a_{12}a_{23}-a_{22}a_{13}   \}$ while the result of the matrix multiplication is 
\begin{align*}
\begin{bmatrix}
    a_{22}  & a_{12}
\end{bmatrix} 
\begin{bmatrix}
    a_{11}       & a_{12} & a_{13} \\
    a_{21}       & a_{22} & a_{23}
\end{bmatrix}
=
\begin{bmatrix}
    a_{11}a_{22}+a_{21}a_{12}  & a_{12}a_{22}+a_{12}a_{22} & a_{22}a_{13}+a_{12}a_{23}
\end{bmatrix}.
\end{align*}
In general, how does one find a map of vector bundles such that its degeneracy locus is the locus of maximal minors of a given matrix? (If possible, it would be nice for the vector bundles to be written as sums of line bundles.)  This seems like an important tool for applying Porteus' formula. 

I believe you got confused with the map. Let's define a bundle map by right multiplication with 
$$
A = \begin{bmatrix}
    P_{11}       & P_{12} & P_{13} \\
    P_{21}       & P_{22} & P_{23}
\end{bmatrix}
$$
whose degrees are respectively$$ \begin{bmatrix}
    a_{11}       & a_{12} & a_{13} \\
    a_{21}       & a_{22} & a_{23}
\end{bmatrix}$$
Start taking $\begin{bmatrix}
    R  & S
\end{bmatrix}$ a section of $\mathcal{O}_{\mathbb{P}^3}(a_{22}) \oplus \mathcal{O}_{\mathbb{P}^3}(a_{12})$. Then 
$$
\begin{bmatrix}
    R  & S
\end{bmatrix} A = \begin{bmatrix}
    RP_{11}+SP_{21}  & RP_{12}+SP_{22} & RP_{13}+SP_{23}
\end{bmatrix} =: C
$$
The identity $a_{1,j}+a_{2,k}=a_{1,k}+a_{2,j}$ says that this map is well defined and the target is $\mathcal{O}_{\mathbb{P}^3}(r) \oplus \mathcal{O}_{\mathbb{P}^3}(s)\oplus \mathcal{O}_{\mathbb{P}^3}(t)$, for some integers $r,s,t$. 
These numbers are the corresponding degrees of the entries of $C$, so $r= a_{22}+a_{11}$, $s= a_{22}+a_{12}$ and $t= a_{22}+a_{13} = a_{12}+a_{23}  $.
Now you want to calculate the points $p\in \mathbb{P}^3$ such that the map $(x,y) \mapsto \begin{bmatrix}
    x  & y
\end{bmatrix}A(p)$  is not injective, the so called degenracy locus. This coincides with the vanishing of the minors of $A(p)$, hence the degeneracy locus of this bundle map is given by vanishing of the minors of $A$.
