Here is a question from the chapter $12$ (on matrices) multiple choice questions in AP Precalculus Premium, $2024$ by Christina Pawlowski-Polanish M.S.:
Multiple-Choice Questions
...
- Calculate the area of the parallelogram with the given vertices $C(0,0)$, $M(2,6)$, $P(11,8)$, and $J(9,2)$.
$\qquad$ (A) $\text{ }100$
$\qquad$ (B) $\text{ }52$
$\qquad$ (C) $\text{ }50$
$\qquad$ (D) $\text{ }49$
Despite having solved this, I'm unsure if my solution is correct or not. Here is the attempt that I have made on this problem:
Now, even though we could just graph this and calculate the area of the parallelogram that way, I think they might want people to solve this using matrices, since it was learned that the absolute value of the determinant of a matrix is equal to the area of the area of its parallelogram. Here is what I did to do that:
Set up a system of equations: For any matrix $$A=\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$We know that one of the vertices can be calculated as$$(a+b,c+d)$$And since we know that that vertice is $$(11,8)$$, we can deduce that that implies$$a=9$$$$b=2$$$$c=2$$$$d=6$$Which we can also imply that because the other $2$ vertices other than $(0,0)$ are calculated as$$(a,b)$$and$$(c,d)$$and therefore, we have the matrix$$A=\begin{bmatrix}9&2\\ 2&6\end{bmatrix}$$Now, to calculate the determinant, we calculate this by$$ad-bc$$$$\implies9\cdot6-2\cdot2=54-4=50$$Therefore, the answer is $50$ square units (since area of shapes are represented (if there is no given measuring unit) in square units) and Option C is correct.
My question
Is the solution that I have come up with correct, or what could I do to attain the correct solution/attain it more easily?
Mistakes I might have made
- Calculating the determinant
- My definition of the area of a parallelogram using matrices
- Attaining the unknown matrix
To clarify
- In all honesty, I don't know a whole lot when it comes to matrices so that might be why my solution would be incorrect if it is.