Finding det(A) with standard basis vectors For example, let $e_1  = [1 \quad 0]$ and $e_2 = [0 \quad 1] $be standard basis vectors. A is a $2 \times 2$ matrix. $Ae_1 = [-3\quad 7]$ and $Ae2 = [3 \quad 5] $ 
How do I find the $\det(A)$?

Since A is a 2x2 matrix, simply use the formula for finding determinants for 2x2 matrices. 
\begin{bmatrix}a&b\\c&d\end{bmatrix}
The formula is ad-bc. In this case, A$e_1$ = \begin{bmatrix}-3\\7\end{bmatrix}
And A$e_2$ = \begin{bmatrix}3\\5\end{bmatrix}
Combine these vectors to get: \begin{bmatrix}-3&3\\7&5\end{bmatrix}
Using the formula, det(A) = (-3 $\cdot$ 5) - (3 $\cdot$ 7) = (-15) - (21) = -36.
