Balancing chemical equations - Matrix determinant is zero for specific reaction

The determinant is zero, because the first two lines are linearly dependent, indeed, we can consider $B_{10}C$ as one "atomic element", or complex, if you like, which provides just one equation,

$B_{10}C:\quad a-2d=0$,

so you need to remove the first line and replace it with the line with $Ni$ instead,

$Ni:\quad b-d=0$.

Then $\det A=4$ and the solution is

$a=4$, $b=2$, $c=12$, $d=2$, $e=4$, $f=12$,

or, equivalently,

$a=2$, $b=1$, $c=6$, $d=1$, $e=2$, $f=6$.

And indeed,

\begin{align} 2B_{10}H_{12}CNH_3 + NiCl_2 + 6NaOH &\rightarrow Na_4(B_{10}H_{10}CNH_2)_2Ni + 2NaCl + 6H_2O . \end{align}