Solve this systems of linear equations with variables $a$, $b$, $c$ and $d$, others are constant. That is, how to solve for $a$, $b$, $c$ and $d$ with a closed form, using the other constant. (To express $a$, $b$, $c$ and $d$ with the others.)
$\left\{\begin{matrix} x_{11}a+x_{21}b+x_{31}c+d=f_1\\ x_{12}a+x_{22}b+x_{32}c+d=f_2\\ x_{13}a+x_{23}b+x_{33}c+d=f_3\\ x_{14}a+x_{24}b+x_{34}c+d=f_4 \end{matrix}\right.$
The solution is a simplex gradient, as defined in some optimization problems.
Also, denote the solution to be $(a,b,c,d)$, can you show that $(x_2-x_1,x_3-x_1,x_4-x_1)^T\cdot(a,b,c)^T=(f_2-f_1,f_3-f_1,f_4-f_1)^T$, where $x_n=(x_{1n},x_{2n},x_{3n})^T$.
I know there are many basic ways to solve for the system but what's the easiest way? And how to show the following equation?
Thanks in advance!
Hint: Assuming $\exists$ a solution to the system, we have: $$\begin{pmatrix}a\\b\\c\\d\end{pmatrix}=\begin{pmatrix}x_{11}&x_{21}&x_{31}&1\\ x_{21}&x_{22}&x_{32}&1\\ x_{13}&x_{21}&x_{33}&1\\ x_{14}&x_{24}&x_{34}&1\end{pmatrix}^{-1}\begin{pmatrix}f_1\\ f_2\\f_3\\f_4\end{pmatrix}$$