An object of area 4cm² is mapped onto an image of area 64cm² under the transformation by matrix((-1, n), (n, 3)). Find the possible values of n.

The area of the object is $4 \text{cm}^2$.
The area of the image is $64 \text{cm}^2$.
The transformation matrix is $M = \begin{pmatrix} n & 2 \\ -1 & n \end{pmatrix}$.
The relationship between the area of the object, the area of the image, and the determinant of the transformation matrix is given by:

$$\text{Area of Image} = |\text{det}(M)| \times \text{Area of Object}$$

For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is $ad - bc$.

$$\text{det}(M) = (n)(n) - (2)(-1)$$ $$\text{det}(M) = n^2 - (-2)$$ $$\text{det}(M) = n^2 + 2$$

Divide both sides by 4:

$$\frac{64}{4} = |n^2 + 2|$$ $$16 = |n^2 + 2|$$

Since $n^2$ is always non-negative, $n^2 + 2$ will always be positive. Therefore, $|n^2 + 2| = n^2 + 2$.

$$16 = n^2 + 2$$

Subtract 2 from both sides:

$$16 - 2 = n^2$$ $$14 = n^2$$

Take the square root of both sides:

$$n = \pm \sqrt{14}$$

The possible values of $n$ are $\sqrt{14}$ and $-\sqrt{14}$.