I'm preparing for an exam currently, and I came across this question: 
I have noticed that A can be constructed from the matrix on the left by a series of row operations, so I had the idea maybe to express A as a product of elementary matrices as well as the matrix on the left and, maybe there was some fact about the determinants of elementary matrices? So then I could just use some properties of determinants, as well as knowing that the determinant of the matrix on the left is 2 to figure out $det(A)$
Entirely confused though because I do not know what the determinant of elementary matrices are, and if I should be multiplying all the elementary matrices together etc etc?
Any help would be hugely appreciated.
Best Answer
Let $B = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$.
If you take the operations that transform $B$ into $A$, and apply them to the identity matrix, you get:
$$C = \begin{pmatrix} 0 & 0 & 4 \\ 0 & 1 & -1 \\ 1 & -2 & 0 \end{pmatrix}$$
So $A = CB$, which means that $\det(A) = \det(C) \det(B)$. You should be able to calculate that $\det(C) = -4$, and you are given that $\det(B) = 2$. Therefore, $\det(A) = -8$.
Since this is nonzero, then yes, $A$ is invertible.