I would like to determine the singular points and the singular values of the polynomial map $f\colon\mathbb{R}^8\to\mathbb{R}^5$ \begin{multline*} f(x_1,\ldots,x_8)=(x_1x_4-x_2x_3,x_5x_8-x_6x_7,x_2x_7-x_3x_6,\\ x_1x_6+x_2(x_8-x_5)-x_4x_6,x_1x_7+x_3(x_8-x_5)-x_4x_7) \end{multline*} motivated by the study of the pairs of matrices $A=\left(\begin{smallmatrix} x_1&x_2\\ x_3&x_4 \end{smallmatrix}\right)$ and $B=\left(\begin{smallmatrix} x_5&x_6\\ x_7&x_8 \end{smallmatrix}\right)$ that commute in $SL(2,\mathbb{R})$: $$AB=BA\Leftrightarrow f(x_1,\ldots,x_8)=(1,1,0,0,0).$$ $$D_{(x_1,\ldots,x_8)}f=\begin{pmatrix} x_4 & -x_3 & -x_2 & x_1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & x_8 & -x_7 & -x_6 & x_5\\ 0 & x_7 & -x_6 & 0 & 0 & -x_3 & x_2 & 0\\ x_6 & (x_8-x_5) & 0 & -x_6 & -x_2 & (x_1-x_4) & 0 & x_2\\ x_7 & 0 & (x_8-x_5) & -x_7 & -x_3 & 0 & (x_1-x_4) & x_3 \end{pmatrix}.$$ $(x_1,\ldots,x_8)$ is a singular point of $f$ if and only if $\mathrm{rank} D_{(x_1,\ldots,x_8)}f<5$. I have to compute $C^3_8=56$ determinants, although the above matrix has some symmytries with respect to $(x_1,\ldots,x_8)$.
Is there a free scientific software (preferably on Linux) or a free platform for calculating determinants (without doing any programming code).
Is there any other techniques to determine the singular points of this map without calculating determinants