Help with linear algebra HW. Please show work!

We have

$$\begin{vmatrix} d & e & f \\ a-2g & b-2h & c-2i \\ 5g & 5h & 5i \end{vmatrix}=5\begin{vmatrix} d & e & f \\ a-2g & b-2h & c-2i \\ g & h & i \end{vmatrix} $$
(Multiply the third row by 2 and add it to the second row. It will not change the determinant)
\[=5\begin{vmatrix} d & e & f \\ a & b & c \\ g & h & i \end{vmatrix}=-5\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \]
\[=\frac{-5}{10}\begin{vmatrix} a & b & c \\ 10d & 10e & 10f \\ g & h & i \end{vmatrix}\]
\[=\frac{-5}{10}\times 7=-\frac{7}{2}.\]
Hence
$$\begin{vmatrix} d & e & f \\ a-2g & b-2h & c-2i \\ 5g & 5h & 5i \end{vmatrix}=-\frac{7}{2}.$$

Note that

1) Switching two rows or columns causes the determinant to switch sign

2) Adding a multiple of one row to another causes the determinant to remain the same

3) Multiplying a *row* as a constant results in the determinant scaling by that constant.