I don't want the answer, but I'd love to kick in the right direction. I'm really not sure how to approach this question.
$$\begin{align}
& -6 = det\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix} \\
& x = det\begin{bmatrix}
a & b & c \\
2d & 2e & 2f \\
g+3a & h+3b & i+3c \\
\end{bmatrix}
& \text{Solve for }x
\end{align}$$
I believe if I set $a=1$, $e=2$, and $i=3$ (all other variables $0$), the determinant of the first matrix is $6$, and then for the second matrix is $12$. These were arbitrary variable initializations and can be any number. The relationship between the two (a scalar multiple of 2) will be the same irrespective of what I set the variables to.
I can then infer that the determinant of the second matrix is $2*det[A]$ or $-12$ because I'm investigating the relationship between the two matrices rather than actually calculating anything. I imagine this is a cheap way out, though. But, from this method I do get $x = -12$, which I believe is the correct answer.
What is the proper way of solving this? I don't want the answer, but I'd like to know the process.
Best Answer
Your answer will be closely related to the $-6$ that was given. You are NOT supposed to find values of $a,b,c,d,...$ that "work". You are supposed to use pure theory.
A) when you add a multiple of one row to another row, the determinant stays the same.
B) multiplying a row by a constant multiplies the determinant by __.