<s>[INST] solve the equation for a: (matrix) = 10 
how did he expand the determinant? 
I tried using the method where you take the determinent by the top left element times the det of the bottom right 4 minus the top element times the det of the bottom left 2 and bottom right 2 + the top right times the det of the bottom left 4 and I got a = 7... am I allowed to use this method to solve it? Did i just miscalculate? how did this person solve this?
 [/INST] $$
\begin{vmatrix} 1 & -1 & 4 \\ -1 & a & 2 \\ 1 & -2 & 3\end{vmatrix} = 1*\begin{vmatrix} a & 2 \\ -2 & 3\end{vmatrix} - (-1)*\begin{vmatrix} -1 & 2 \\ 1 & 3\end{vmatrix} + 4*\begin{vmatrix} -1 & a \\ 1 & -2\end{vmatrix} = 10
$$
can you take it from here?
Also use:
$$
\begin{vmatrix} a & b \\ c & d\end{vmatrix} = ad-bc.
$$
note this is just to verify the OP answer.
$$
1*(3a+4) + (-3 - 2) + 4*(2-a) = 3a + 4 - 5 + 8 - 4a = -a +7 = 10
$$
this leads to a = -3
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