Non-bashy Matrix Multiplication

Question

MathStack Exchange professional people,

I had a question about a problem that I was working on for my pre-calculus class.

Here's the problem statement:

The area of the parallelogram with vertices ${0}$, ${v}$, ${w}$, and ${v} + {w}$ is 34. Find the area of the parallelogram with vertices ${0}$, ${A} {v}$, ${A} {w}$, and ${A} {v} + {A} {w}$, where $${A} = \begin{pmatrix} 3 & -5 \\ -1 & -3 \end{pmatrix}.$$

I got the answer by doing something very tedious. I set $v=\binom{17}{0}$ and $w=\binom{0}{2}$, and did some really crazy matrix multiplication and a lot of plotting points of GeoGebra to get the answer of: $\boxed{476}$.

Now, I'm 100% sure that was not the fastest way, can someone tell me the non-bash way to do the problem?

Thanks, NoobMathist

Answer 1

Hint: If $v=\begin{pmatrix}a\\b\end{pmatrix}$ and $w=\begin{pmatrix}c\\d\end{pmatrix}$ then the area of the original parallelogram is $$\left|\det \begin{pmatrix}a & b \\ c & d\end{pmatrix}\right| = |ad-bc|.$$

Provide a similar expression for the area of the second parallelogram, and relate it to the above expression using properties of determinants.

Answer 2

You're probably supposed to know that the absolute value of the determinant of $A$ is the area scale factor of the linear transformation represented by the matrix $A$. With this knowledge, you can easily solve the problem in your head.