how to determine rows making determinant of matrix nearly singular

Question

I have a 10*10 matrix, whose determinant is -3.8985e-07 (approximately zero). When I reduce it echelon form, it shows me complete rank. I am not sure which row is causing it to be approximately singular.

My matrix is as follow: A = \begin{matrix} 0.0426 & 0.0032 & 0.0029 & 0.0264 & -0.0094 & 0.3868 & -0.1292 & 0.1259 & 0.0417 & 0.3669 \\ 0.0007 & 0.0006 & -0.0121 & -0.0594 & 0.0142 & 0.2541 & 0.4486 & 0.0034 & -0.5867 & -0.0168 \\ -0.0010 & 0.0061 & -0.0297 & -0.1211 & 0.0662 & 0.1349 & -0.2580 & 0.0180 & -0.2322 & 0.0887 \\ -0.0027 & 0.0170 & -0.0838 & -0.3270 & 0.2099 & 0.2917 & -0.7196 & -0.5922 & 2.7905 & 0.5151 \\ 0.0001 & -0.0004 & 0.0019 & 0.0078 & -0.0048 & -0.0158 & 0.0212 & 0.0410 & -0.0650 & 5.6639 \\ 0.0010 & 0.0002 & -0.0203 & -0.0481 & 0.0529 & -1.1830 & 0.4618 & 3.6336 & 0.5938 & -2.3641 \\ 0.0141 & 0.0036 & -0.0024 & 0.0187 & 0.0117 & -0.0099 & -0.0129 & 0.0097 & 0.0079 & 0.0031 \\ -0.0006 & 0.0016 & -0.0010 & 0.0005 & 0.0124 & -0.0010 & 0.0164 & 0.0137 & -0.0221 & -0.0110\\ 0.0096 & -0.0162 & -0.0020 & -0.0046 & -0.0012 & -0.0026 & -0.0002 & -0.0008 & -0.0001 & 0.0001\\ -0.0411 & 0.0951 & -0.0257 & 0.0668 & 0.8981 & -1.8226 & -2.7494 & 0.9926 & 5.6168 & -0.7285 \end{matrix}

The above matrix is generated by the following procedure:

A = [JEE*M^{-1};  
     JOr*M^{-1};  
     JCOM*M^{-1};  
     M^{-1}(9,:)]

Value of JEE, JOr, JCOM and M^{-1} are given on link https://pastebin.com/dmRP4xKj

JEE = \begin{matrix} 0.4116 & 0.0509 & 0.1475 & 0.3754 & 0.0214 & 0.3278 & -0.0762 & 0.1872 & 0.0406 & 0\\ 0 & -0.0000 & -0.3059 & -0.0131 & 0.2994 & 0.0997 & 0.2411 & -0.0044 & -0.0024 & 0 \\ -0.1073 & -0.4639 & -0.2431 & -0.0589 & -0.0014 & -0.0507 & -0.2808 & 0.0389 & -0.1958 & 0 \end{matrix}

Jor = \begin{matrix} 0 & 0 & -0.8549 & -0.0276 & 0.6819 & 0.3221 & 0.9272 & -0.1869 & 0.9788 & 0.0248 \\ 1.0000 & 1.0000 & -0.0000 & -0.9986 & -0.0521 & -0.8741 & 0.3688 & 0.3049 & -0.0228 & 0.9997 \\ -0.0000 & -0.0000 & -0.5187 & 0.0455 & -0.7296 & 0.3635 & 0.0652 & 0.9339 & 0.2033 & -0.0072 \end{matrix}

JCOM = \begin{matrix} 0.1918 & 0.0938 & 0.0407 & 0.0265 & 0.0182 & 0.0158 & -0.0044 & 0.0057 & 0.0012 & 0 \\ 0 & 0 & -0.0208 & -0.0012 & 0.0260 & 0.0053 & 0.0129 & -0.0002 & -0.0001 & 0 \\ 0.0337 & -0.1232 & -0.0670 & -0.0104 & 0.0151 & -0.0012 & -0.0107 & 0.0012 & -0.0059 & 0 \end{matrix}

M_inverse = \begin{matrix} 0.1093 & -0.0904 & 0.0477 & -0.1160 & -0.0937 & 0.1243 & 0.1222 & -0.0806 & -0.0720 & -0.0295 \\ -0.0765 & 0.2780 & -0.3772 & 0.6788 & 0.0228 & -0.5932 & -0.2216 & 0.0015 & 0.0951 & 0.0558 \\ 0.0382 & -0.3772 & 1.3700 & -3.4682 & 0.3792 & 3.2918 & -0.3330 & -0.0489 & -0.0257 & -0.0975 \\ -0.0428 & 0.6788 & -3.4682 & 12.9593 & -0.4682 & -13.8475 & 2.4026 & 2.2228 & 0.0668 & -1.3056 \\ -0.0478 & 0.0228 & 0.3792 & -0.4682 & 1.3084 & -0.7536 & -0.7630 & 1.4545 & 0.8981 & -1.1341 \\ 0.0857 & -0.5932 & 3.2918 & -13.8475 & -0.7536 & 18.6303 & -2.4133 & -5.9961 & -1.8226 & 5.5496 \\ 0.0717 & -0.2216 & -0.3330 & 2.4026 & -0.7630 & -2.4133 & 3.5014 & 0.8804 & -2.7494 & -1.2524 \\ -0.0426 & 0.0015 & -0.0489 & 2.2228 & 1.4545 & -5.9961 & 0.8804 & 6.9075 & 0.9926 & -5.2356 \\ -0.0411 & 0.0951 & -0.0257 & 0.0668 & 0.8981 & -1.8226 & -2.7494 & 0.9926 & 5.6168 & -0.7285 \\ -0.0174 & 0.0558 & -0.0975 & -1.3056 & -1.1341 & 5.5496 & -1.2524 & -5.2356 & -0.7285 & 11.1710 \end{matrix}

Any guidance will be highly appreciated.

Answer 1

I think I found the solution. The problem is coming from JCOM*M^{-1} term. I tried finding the pseudo inverse of every term and determinant of $\left(JCOM*M^{-1}\right)*\left(JCOM*M^{-1}^{T}\right)^{T}$ has determinant very close to zero. So I believe the near singularity must be coming from it.