Linear algebra calculus trick. I have a matrix and a vector:
$$
A=\begin{bmatrix}
a &b\\
c&d
\end{bmatrix},
$$
$$
\vec v=\begin{bmatrix}
a+b\\
c+d
\end{bmatrix}
$$
Is there an algebraic operation that produce the following matrix:
$$
B=\begin{bmatrix}
\dfrac{a}{a+b} &\dfrac{b}{a+b}\\
\dfrac{c}{c+d} &\dfrac{d}{c+d}
\end{bmatrix}?
$$
 A: I'm not really sure what you'll consider an "algebraic operation", but I guess there's
$$ \left(
\left[\begin{matrix} 1 & 0 \end{matrix}\right]
\vec v
\left[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right]
+
\left[\begin{matrix} 0 & 1 \end{matrix}\right]
\vec v
\left[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}\right]
\right)^{-1}
A
$$
A: Inspired by Steven's solution, I have the following one which is a generalization. 
As I understand from your matrix, I assume that you want to do the following:
$$
A=\begin{pmatrix}
a_{11}& \cdots&a_{1n}\\
a_{21}& \cdots&a_{2n}\\
\vdots&\ddots&\vdots\\
a_{n1}& \cdots&a_{nn}\\
\end{pmatrix},
$$
and
$$
b=\begin{pmatrix}
\sum_{i=1}^{n}a_{1i}\\
\sum_{i=1}^{n}a_{2i}\\
\vdots\\
\sum_{i=1}^{n}a_{ni}\\\\
\end{pmatrix}.
$$
Find the following matrix $B$?
$$
B=\begin{pmatrix}
\dfrac{a_{11}}{\sum_{i=1}^{n}a_{1i}}& \cdots&\dfrac{a_{1n}}{\sum_{i=1}^{n}a_{1i}}\\
\dfrac{a_{21}}{\sum_{i=1}^{n}a_{2i}}& \cdots&\dfrac{a_{2n}}{\sum_{i=1}^{n}a_{2i}}\\
\vdots&\ddots&\vdots\\
\dfrac{a_{n1}}{\sum_{i=1}^{n}a_{ni}}& \cdots&\dfrac{a_{nn}}{\sum_{i=1}^{n}a_{ni}}\\
\end{pmatrix},
$$
Let $C$ be the following matrix, $C=\mathrm{diag}(b)= \begin{pmatrix}
   \sum_{i=1}^{n}a_{1i}&&&&\huge0\\
    & & \ddots\\
\huge0    & & & & \sum_{i=1}^{n}a_{ni}
 \end{pmatrix},$
Therefore,
$$B=C^{-1}A.$$

In your problem, 
$$
B=\begin{bmatrix}
\dfrac{a}{a+b} &\dfrac{b}{a+b}\\
\dfrac{c}{c+d} &\dfrac{d}{c+d}
\end{bmatrix}=\mathrm{diag}(b)^{-1}\cdot A=\begin{bmatrix}
\dfrac{1}{a+b} &0\\
0 &\dfrac{1}{c+d}
\end{bmatrix}\cdot\begin{bmatrix}
a &b\\
c &d
\end{bmatrix}
$$
