Question about null space balancing method

Question

I'm trying to use the null space method to balance the following equation: . I obtained the following composition matrix: $$\begin{bmatrix} 3 & 8 & 1 & 12 & 4 & 2 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 4 & 4 & 3 & 40 & 3 & 1 \\ 0 & 2 & 1 & 3 & 1 & 0 \\ 0 & 1 & 0 & 12 & 0 & 0 \end{bmatrix}$$ where rows are in order H P O N Mo. I take the rref of this matrix, and augment it with one row of zeroes except the last element is 1. Taking the inverse of that matrix, I get $$\begin{bmatrix} \frac{1}{51} & \frac{4}{17} & \frac{35}{51} & -\frac{1}{51} & -\frac{56}{51} & 1\\ \end{bmatrix}.$$ After scaling the elements by 51 I would get the end result of coefficients: 1, 12, 35, 1, 56, 51 which does not at all balance the equation. A correct solution would be 1, 12, 21, 1, 21, 12.

I have used the exact same steps to balance other equations. What am I doing wrong?

Answer 1

Fortunately this issue was just caused by a small oversight. In the midst of so many coefficients, you accidentally skipped the second nitrogen atom in $\ce{NH4NO3}$, which changes the fifth element in the fourth row from 1 to 2. The correct composition matrix is then:

$$\begin{bmatrix} 3 & 8 & 1 & 12 & 4 & 2 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 4 & 4 & 3 & 40 & 3 & 1 \\ 0 & 2 & 1 & 3 & \color{red}{\textbf{2}} & 0 \\ 0 & 1 & 0 & 12 & 0 & 0 \end{bmatrix}$$

After complementing with the bottom row $\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}$ and inverting the matrix, we obtain:

$$\frac{1}{96}\begin{bmatrix} 3 & 79 & 2 & -9 & -14 & -8 \\ 36 & -204 & 24 & -108 & -72 & -96 \\ 39 & -477 & 90 & -213 & -246 & -168 \\ -3 & 17 & -2 & 9 & 14 & 8 \\ -51 & 417 & -66 & 249 & 174 & 168 \\ 0 & 0 & 0 & 0 & 0 & 96 \end{bmatrix}$$

Finally, transposing the rightmost column and dividing its coefficients by 8, the result is $\begin{bmatrix} -1 & -12 & -21 & 1 & 21 & 12 \end{bmatrix}$, exactly as expected.

Thank you for the linked article! Hopefully others may find it useful. If you're into linear algebra applied to chemistry, I recommend you take a peek into chemical reaction network theory!