Find the determinants below using the fact: $\det \left[\begin{smallmatrix}a&b&c\\d&e&f\\g&h&i\end{smallmatrix}\right]=7$ a. \begin{bmatrix}g&h&i\\2d&2e&2f\\3a&3b&3c\end{bmatrix}
b. \begin{bmatrix}a&b&c\\d-2a&e-2b&f-2c\\5g&5h&5i\end{bmatrix}
Hello, I am not sure how to go about answering this question. I don't need and exact answer, but I just need to know how to get started on answering part A and B of this question.
 A: Hint: 
a)
\begin{align}
\begin{vmatrix}
a & b & c \\
2d & 2e & 2f\\
3g & 3h  & 3i
\end{vmatrix}
=2\begin{vmatrix}
a & b & c \\
d & e & f\\
3g & 3h  & 3i
\end{vmatrix}
\end{align}
b)
\begin{align}
\begin{vmatrix}
a & b & c \\
d-a & e-b & f-c\\
5g & 5h  & 5i
\end{vmatrix}
=\begin{vmatrix}
a & b & c \\
d & e & f\\
5g & 5h  & 5i
\end{vmatrix}
\end{align}
A: Lets try solve a similar determinant:
$$\det\begin{pmatrix}2d & 2e & 2f \\ a-3g & b - 3h & c - 3i \\4g & 4h & 4i\end{pmatrix} $$
We have that:
\begin{align*}
\det\begin{pmatrix}2d & 2e & 2f \\ a-3g & b - 3h & c - 3i \\4g & 4h & 4i\end{pmatrix} & = 
2\det\begin{pmatrix}d & e & f \\ a-3g & b - 3h & c - 3i \\4g & 4h & 4i\end{pmatrix} \\
& \text{Multiply the first row by a constant} \\
& = 8\det\begin{pmatrix}d & e & f \\ a-3g & b - 3h & c - 3i \\g & h & i\end{pmatrix} \\
& \text{Multiply the third row by a constant} \\
& = 8\det\begin{pmatrix}d & e & f \\ a & b  & c  \\g & h & i\end{pmatrix} \\
&\text{Add three times the third row to the second row} \\
& = -8\det\begin{pmatrix} a & b  & c  \\d & e & f \\g & h & i\end{pmatrix} \\
&\text{Switch the first and second rows} \\
& = -8\times (7) = -56 \\
\text{Substitute in the determinant we know}
\end{align*}
Hopefully you see how you can solve your problems from this.
A: Use the property that $\det(\mathrm{AB}) = \det(\mathrm{A})\det(\mathrm{B})$
Decompose your given matrices into a product of row-operation matrices with your given matrix.  For instance:
$$\begin{bmatrix}a&b&c\\2d&2e&2f\\g&h&i\end{bmatrix} = 
  \begin{bmatrix}1&0&0\\0&2&0\\0&0&1\end{bmatrix}
  \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$$
So that matrix's determinant is 14.
